nck_approx_plots.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#from fractions import Fraction\n",
    "#from math import log\n",
    "import matplotlib.pyplot as plt\n",
    "#import numpy as np\n",
    "import sympy as sp\n",
    "\n",
    "sp.init_printing()\n",
    "n, i = sp.symbols(\"n i\", integer=True, positive=True)\n",
    "k = sp.symbols(\"k\", integer=True, nonnegative=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# True Function\n",
    "def nCk(n,k):\n",
    "    result=1\n",
    "    k = min(k,n-k)\n",
    "    return int(np.prod(\n",
    "        (np.arange(n,n-k,-1) + Fraction())\n",
    "        / np.arange(1,k+1)\n",
    "    ))\n",
    "\n",
    "def nCk_iter(n):\n",
    "    result = 1\n",
    "    i = 0\n",
    "    while True:\n",
    "        yield result\n",
    "        if i >= n: break\n",
    "        result = (result*Fraction(n-i, i+1))._numerator\n",
    "        i += 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hyper_sq(x):\n",
    "    return x ** x\n",
    "\n",
    "_1div2 = sp.Rational(1,2)\n",
    "\n",
    "nCk_ests_sp = [\n",
    "    # Trivial estimate\n",
    "    n ** k,\n",
    "    # Simple estimate\n",
    "    (n-k+1) ** k,\n",
    "    # Gauss approx.\n",
    "    n ** (k*(n-k)/(n-1)),\n",
    "    # Integral approx.\n",
    "    sp.sqrt(_1div2)* hyper_sq(n+_1div2) / (\n",
    "        hyper_sq(n-k+_1div2) * hyper_sq(k+_1div2))\n",
    "]\n",
    "for j in range(len(nCk_ests_sp)):\n",
    "    nCk_ests_sp[j] = nCk_ests_sp[j].simplify()\n",
    "del _1div2\n",
    "\n",
    "est_labels = (\n",
    "    \"trivial\",\n",
    "    \"simple\",\n",
    "    \"Gaussian approx.\",\n",
    "    \"Integral approx.\"\n",
    ")\n",
    "\n",
    "est_styles = (\n",
    "    \"m\",\n",
    "    \"r\",\n",
    "    \"y\",\n",
    "    \"g\",    \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_nCk_w_est(indices_ests):\n",
    "    for n_val in (3,10,100):\n",
    "        plot = sp.plotting.plot(\n",
    "            sp.binomial(n_val, k), (k, 0, n_val),\n",
    "            line_color=\"k\",\n",
    "            title=\"nCk & Estimates (n = \" + str(n_val) + \")\",\n",
    "            xlabel=\"k-values\",\n",
    "            xscale=\"linear\",\n",
    "            yscale=\"log\",\n",
    "            #axis_center=(0,1), #causes strange errors in plot display\n",
    "            #legend=True,\n",
    "            show=False\n",
    "        )\n",
    "        \n",
    "        for index in indices_ests:\n",
    "            plot.extend(sp.plotting.plot(\n",
    "                nCk_ests_sp[index].subs(n,n_val), (k, 0, n_val),\n",
    "                line_color=est_styles[index],\n",
    "                title=est_labels[index],\n",
    "                show=False\n",
    "            ))\n",
    "\n",
    "        plot.show()\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Proofs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAaCAYAAABGiCfwAAAABHNCSVQICAgIfAhkiAAAAfFJREFUSInt1UuIzlEYx/HPO4Yat3Gp0WCSMAxySbkUC1MWsrG0GYysNMnEwo6FhcKGshKys2Ix5dJsbFxCpNyymIyFaHIZxsIoFue89e8///PO+9a7oPw25/+c/3me73Nuz+Ev0nJsxyz04jo2ohuHcKbAZzouYGqtoKtoxH5MxBvsygQdSfhuQF/0HVdT8ATzoz0N8zCYGbMVDyvEOIHj1cBOxcFZ7cHljH0eR9CMUkGMFnyJSSY1G8OYm+u/gt3xu4QhYeaHK8S6hJOVYAdwv6D/Llozdh/2CfuTUhfeKp45uIGzlbKpQQvxGx3QkPtZwiY8qhNsAN+wvgjWKmz4uzrB4BMWp2DwtY6wz8KhGwObHNvhOsK+YxJjb/jPHDSvFhyMfkerhDXhB2NnNhTbmQnHj0LVWFElqBzrfRFsEKNC4U1pDfqrBDWgDa+LYKN4itUVAnTWAFsqFPB7RTC4hc0J5yYswHPswGnhHk1IjN+Cx+IyFqkjBig6JNtwG3uFAtsoVImU+tFTNopm9hJ3hDcsr078iu2q+D2QAC3BMlyskAxoxythvbN6IDyq7cJlhZWJGNewczxQWV04l7Gb8UGon3PwIvb3Fvh241i+M7Wx8Ex4rxbFwOswI2Y8IlyBNtwU6l82qbXC4fmvf1x/AK4LU3UVdQPmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$${\\binom{n}{k}}$$"
      ],
      "text/plain": [
       "⎛n⎞\n",
       "⎜ ⎟\n",
       "⎝k⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nCk = sp.binomial(n,k)\n",
    "display(nCk)\n",
    "\n",
    "#nCk_ln = sp.Sum(sp.ln(n-i+1) - sp.ln(i), (i, 1, k))\n",
    "#display(nCk_ln)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Proof - for $n\\hat{C}k = n^{\\left(\\frac{k(n-k)}{n-1}\\right)}$\n",
    "\n",
    "Goal: WTS $nCk = O\\left(n\\hat{C}k\\right)$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff4f684d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50d86da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff4f29ab00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_nCk_w_est([2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$n^{- \\frac{k \\left(k - n\\right)}{n - 1}}$$"
      ],
      "text/plain": [
       " -k⋅(k - n) \n",
       " ───────────\n",
       "    n - 1   \n",
       "n           "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nCk_est = nCk_ests_sp[2]\n",
    "\n",
    "nCk_est"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rephrasing Goal\n",
    "- $nCk = nC(n-k)$, $n\\hat{C}k = n\\hat{C}(n-k)$\n",
    "- Small $n,k$ cases - $0C0 = 0\\hat{C}0 = 1$\n",
    "- ETS that $\\forall n\\in\\mathbb{N}$, $k\\in [0,\\frac{n}{2}]$: $nCk \\leq n\\hat{C}k$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nCk_est is symmetric about k = n/2\n",
      "0C0_est = 1\n"
     ]
    }
   ],
   "source": [
    "assert nCk_est == nCk_est.subs(k, n-k).simplify()\n",
    "print(\"nCk_est is symmetric about k = n/2\")\n",
    "\n",
    "assert nCk_est.subs({n: 0, k: 0}).simplify() == 1\n",
    "print(\"0C0_est = 1\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Element-wise increase\n",
    "- $nCk = \\prod_{i=1}^{k} \\frac{n+1}{i}-1$\n",
    "- $n\\hat{C}k = n^{\\left(\\frac{k(n-k)}{n-1}\\right)}\n",
    "    = \\prod_{i=1}^{k} {\\left(n^\\frac1{n-1}\\right)} ^ {i(n-i) - (i-1)(n-i+1)}\n",
    "    = \\prod_{i=1}^{k} {\\left(n^\\frac1{n-1}\\right)} ^ {n+1-2i}$\n",
    "- Let $f_n(i) := \\log_n\\left( \\frac{n\\hat{C}i}{n\\hat{C}(i-1)} \\cdot \\frac{nC(i-1)}{nCi} \\right)\n",
    "    = \\frac{n+1-2i}{n-1} + \\log_n(i) - \\log_n(n-i+1)$\n",
    "- ETS that $f_n(i) \\geq 0\\ \\forall n\\in\\mathbb{N}$, $i\\in [1,\\frac{n}{2}]$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{2 i}{n - 1} + \\frac{n}{n - 1} + \\frac{\\log{\\left (i \\right )}}{\\log{\\left (n \\right )}} + \\frac{1}{\\log{\\left (n \\right )}} \\log{\\left (\\frac{1}{- i + n + 1} \\right )} + \\frac{1}{n - 1}$$"
      ],
      "text/plain": [
       "                              ⎛    1     ⎞        \n",
       "                           log⎜──────────⎟        \n",
       "   2⋅i      n     log(i)      ⎝-i + n + 1⎠     1  \n",
       "- ───── + ───── + ────── + ─────────────── + ─────\n",
       "  n - 1   n - 1   log(n)        log(n)       n - 1"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sp.log(\n",
    "    (nCk_est.subs(k,i) / nCk_est.subs(k,i-1))\n",
    "    / (nCk.subs(k,i) / nCk.subs(k,i-1)),\n",
    "    n)\n",
    "f = sp.simplify(f)\n",
    "\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivatives of $f$:\n",
    "- $\\frac{d}{di} f_n(i) = \\frac{-2}{n-1} + \\frac1{\\ln(n)i} + \\frac1{\\ln(n)(n+1-i)}\n",
    "    = \\frac{-2}{n-1} + \\frac{n+1}{i(n+1-i)\\ln(n)}$\n",
    "- $\\frac{d^2}{di^2} f_n(i) = 0 + \\frac{-1}{\\ln(n)i^2} + \\frac1{\\ln(n)(n+1-i)^2}\n",
    "    = \\frac1{\\ln(n)} \\left( {\\frac1{n+1-i}}^2 - {\\frac1{i}}^2 \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First and second derivatives are as derived above\n"
     ]
    }
   ],
   "source": [
    "der_f_comp = sp.diff(f, i)\n",
    "der2_f_comp = sp.diff(der_f_comp, i)\n",
    "#display(der_f_comp)\n",
    "#display(der2_f_comp)\n",
    "\n",
    "der_f_byhand = (-2 / (n-1)) + ((n+1) / (i*(n+1-i)*sp.log(n)))\n",
    "der2_f_byhand = ((1/(n+1-i)**2) - (1/i**2)) / sp.log(n)\n",
    "\n",
    "n_val = 1000\n",
    "assert sp.simplify(der_f_comp - der_f_byhand) == 0\n",
    "#sp.plotting.plot(der_f_comp.subs(n,n_val) - der_f_byhand.subs(n,n_val), (i,1,n_val))\n",
    "assert sp.simplify(der2_f_comp - der2_f_byhand) == 0\n",
    "#sp.plotting.plot(der2_f_comp.subs(n,n_val) - der2_f_byhand.subs(n,n_val), (i,1,n_val))\n",
    "print(\"First and second derivatives are as derived above\")\n",
    "\n",
    "der_f = der_f_byhand\n",
    "der2_f = der2_f_byhand\n",
    "del der_f_byhand, der2_f_byhand, der_f_comp, der2_f_comp\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proof by Convexity\n",
    "- for $i\\in[1,\\frac{n+1}{2}]$, $i \\leq n+1-i \\Rightarrow {\\frac1{n+1-i}}^2 - {\\frac1{i}}^2 \\leq 0$\n",
    "    - $\\Rightarrow \\frac{d^2}{di^2} f_n(i) \\leq 0$\n",
    "- $f_n(1) = \\frac{n+1-2}{n-1} + \\log_n(1) - \\log_n(n-1+1) = \\frac{1}{1} - 1 = 0$\n",
    "- $f_n\\left(\\frac{n+1}{2}\\right)\n",
    "    = \\frac{0}{n-1} + \\log_n\\left(\\frac{\\frac{n+1}{2}}{\\frac{n+1}{2}}\\right) = 0$\n",
    "- $\\Rightarrow \\forall i\\in[1,\\frac{n+1}{2}]$, $f_n(i) \\geq 0$\n",
    "- $\\therefore nCk = O\\left(n\\hat{C}k\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Demo: Assertion of Convexity\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff4f2f8860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff4f20dbe0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff4f32b630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50e23d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# TODO make a cleaner demo of convexity\n",
    "n_val = 1000\n",
    "i_last = (n_val+1)/2\n",
    "\n",
    "print(\"Demo: Assertion of Convexity\")\n",
    "for i_first in (1, 100, 450, 500):\n",
    "    sp.plotting.plot(der2_f.subs(n,n_val).simplify(), (i, i_first, i_last),\n",
    "                     axis_center = (i_last,0), title = \"$\\\\frac{d^2}{di^2} f < 0$ for $i \\\\leq \\\\frac{n+1}{2}$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Assertion of Endpoints\n",
      "\tf_n(1) = 0\n",
      "\tf_n((n+1)/2) = 0\n"
     ]
    }
   ],
   "source": [
    "print(\"Assertion of Endpoints\")\n",
    "\n",
    "assert f.subs(i, 1).simplify() == 0\n",
    "print(\"\\tf_n(1) = 0\")\n",
    "\n",
    "assert f.subs(i, (n+1)/2).simplify() == 0\n",
    "print(\"\\tf_n((n+1)/2) = 0\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### QED"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Proof - for $n\\hat{C}k = \\sqrt{\\frac{1}{2}} \\cdot \\frac{^2 \\left(n+\\frac{1}{2}\\right)}{\n",
    "        ^2 \\left(n-k+\\frac{1}{2}\\right)\n",
    "        \\cdot ^2 \\left(k+\\frac{1}{2}\\right)\n",
    "        }$\n",
    "\n",
    "*Formulation* - $\\ln{\\left(n\\hat{C}k\\right)}\n",
    "    := \\int_{\\frac12}^{k-\\frac12} \\left[\\ln{(n+1-x)}-\\ln{(x)}\\right]dx$\n",
    "\n",
    "***Goal***: WTS $nCk = \\Theta\\left(n\\hat{C}k\\right)$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff512b8080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff513711d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff513027b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_nCk_w_est([3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(2 k + 1\\right)^{- k - \\frac{1}{2}} \\left(2 n + 1\\right)^{n + \\frac{1}{2}} \\left(- 2 k + 2 n + 1\\right)^{k - n - \\frac{1}{2}}$$"
      ],
      "text/plain": [
       "         -k - 1/2          n + 1/2                 k - n - 1/2\n",
       "(2⋅k + 1)        ⋅(2⋅n + 1)       ⋅(-2⋅k + 2⋅n + 1)           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(- k - \\frac{1}{2}\\right) \\log{\\left (2 k + 1 \\right )} + \\left(n + \\frac{1}{2}\\right) \\log{\\left (2 n + 1 \\right )} + \\log{\\left (\\left(- 2 k + 2 n + 1\\right)^{k - n - \\frac{1}{2}} \\right )}$$"
      ],
      "text/plain": [
       "                                                      ⎛                k - n -\n",
       "(-k - 1/2)⋅log(2⋅k + 1) + (n + 1/2)⋅log(2⋅n + 1) + log⎝(-2⋅k + 2⋅n + 1)       \n",
       "\n",
       " 1/2⎞\n",
       "    ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nCk_est = nCk_ests_sp[3]\n",
    "nCk_est_ln = sp.expand_log(sp.ln(nCk_est))\n",
    "\n",
    "display(nCk_est)\n",
    "display(nCk_est_ln)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### First: WTS $nCk = O\\left(n\\hat{C}k\\right)$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Rephrasing Goals\n",
    "- $nCk = nC(n-k)$, $n\\hat{C}k = n\\hat{C}(n-k)$\n",
    "- ETS that $\\forall n\\in\\mathbb{Z}_+$, $k\\in\\left[0,\\frac{n}{2}\\right]$: $nCk \\leq n\\hat{C}k$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nCk_est is symmetric about k = n/2\n"
     ]
    }
   ],
   "source": [
    "assert nCk_est == nCk_est.subs(k,n-k).simplify()\n",
    "print(\"nCk_est is symmetric about k = n/2\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Element-wise increase\n",
    "- $nCk = \\prod_{i=1}^{k} \\frac{n+1}{i}-1$\n",
    "- $n\\hat{C}k = \\sqrt{\\frac{1}{2}} \\cdot \\frac{^2 (n+0.5)}{^2 (n-k+0.5) \\cdot ^2 (k+0.5)}\n",
    "    = \\prod_{i=1}^{k} \\frac{^2 (n-k+1.5) \\cdot ^2 (k-0.5)}{^2 (n-k+0.5) \\cdot ^2 (k+0.5)}$\n",
    "- Let $f_n(i) := \\ln\\left( \\frac{n\\hat{C}k}{n\\hat{C}(k-1)} \\cdot \\frac{nC(k-1)}{nCk} \\right)$\n",
    "    - (Let $xlnx(i) := i\\ln(i)$)\n",
    "    - $= \\left[xlnx(n-i+1.5) + xlnx(i-0.5) - xlnx(n-i+0.5) - xlnx(i+0.5)\\right]\n",
    "        - \\ln(n-i+1) + \\ln(i)$\n",
    "- ETS that $f_n(i) \\geq 0\\ \\ \\forall n\\in\\mathbb{N}$, $k\\in [1,\\frac{n}{2}]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FURRFUezyfxPrAqLZZPu6AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(- i - \\frac{1}{2}\\right) \\log{\\left (2 i + 1 \\right )} + \\left(i - \\frac{1}{2}\\right) \\log{\\left (2 i - 1 \\right )} + \\left(- i + n + \\frac{3}{2}\\right) \\log{\\left (- 2 i + 2 n + 3 \\right )} + \\left(i - n - \\frac{1}{2}\\right) \\log{\\left (- 2 i + 2 n + 1 \\right )} + \\log{\\left (i \\right )} - \\log{\\left (- i + n + 1 \\right )}$$"
      ],
      "text/plain": [
       "(-i - 1/2)⋅log(2⋅i + 1) + (i - 1/2)⋅log(2⋅i - 1) + (-i + n + 3/2)⋅log(-2⋅i + 2\n",
       "⋅n + 3) + (i - n - 1/2)⋅log(-2⋅i + 2⋅n + 1) + log(i) - log(-i + n + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff512d9828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7eff51098940>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sp.log(\n",
    "    (nCk_est.subs(k,i) / nCk_est.subs(k,i-1)).simplify()\n",
    "    / (nCk.subs(k,i) / nCk.subs(k,i-1)).simplify()\n",
    ")\n",
    "f = sp.expand_log(f, force=True)#.refine(sp.Q.nonnegative(-i+n)))\n",
    "\n",
    "display(f)\n",
    "n_val = 1000\n",
    "sp.plotting.plot(f.subs(n,n_val), (i, 1, n_val))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Derivatives of $f$\n",
    "- $\\frac{d}{di} f_n(i)\n",
    "    = \\frac{n+1}{i(n+1-i)} -\n",
    "    ( \\ln{(2i+1)}\n",
    "    - \\ln{(2i-1)}\n",
    "    + \\ln{(2(n-i)+3)}\n",
    "    - \\ln{(2(n-i)+1)})$\n",
    "\n",
    "- $\\frac{d^2}{di^2} f_n(i) = \\frac\n",
    "    {(n+1)(n+1-2i)(4n^2 - 8(n-i)(i-1) + 3)}\n",
    "    {i^2 (2i-1)(2i+1) (n+1-i)^2 (2n+3-2i) (2n+1-2i)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\log{\\left (2 i - 1 \\right )} - \\log{\\left (2 i + 1 \\right )} + \\log{\\left (- 2 i + 2 n + 1 \\right )} - \\log{\\left (- 2 i + 2 n + 3 \\right )} + \\frac{n + 1}{i \\left(- i + n + 1\\right)}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "log(2⋅i - 1) - log(2⋅i + 1) + log(-2⋅i + 2⋅n + 1) - log(-2⋅i + 2⋅n + 3) + ────\n",
       "                                                                          i⋅(-\n",
       "\n",
       "n + 1     \n",
       "──────────\n",
       "i + n + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff51325390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\left(n + 1\\right) \\left(- 2 i + n + 1\\right) \\left(4 n^{2} - \\left(- 8 i + 8 n\\right) \\left(i - 1\\right) + 3\\right)}{i^{2} \\left(2 i - 1\\right) \\left(2 i + 1\\right) \\left(- 2 i + 2 n + 1\\right) \\left(- 2 i + 2 n + 3\\right) \\left(- i + n + 1\\right)^{2}}$$"
      ],
      "text/plain": [
       "                              ⎛   2                           ⎞       \n",
       "       (n + 1)⋅(-2⋅i + n + 1)⋅⎝4⋅n  - (-8⋅i + 8⋅n)⋅(i - 1) + 3⎠       \n",
       "──────────────────────────────────────────────────────────────────────\n",
       " 2                                                                   2\n",
       "i ⋅(2⋅i - 1)⋅(2⋅i + 1)⋅(-2⋅i + 2⋅n + 1)⋅(-2⋅i + 2⋅n + 3)⋅(-i + n + 1) "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff5151acc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First and second derivatives are as derived above\n"
     ]
    }
   ],
   "source": [
    "der_f_comp = sp.expand_log(f.diff(i).simplify())#.refine(sp.Q.nonnegative(-i+n)))\n",
    "der2_f_comp = sp.diff(der_f_comp, i).factor()\n",
    "\n",
    "der_f_byhand = ((n+1)/(i*(n+1-i))) - (\n",
    "    sp.ln(2*i+1) - sp.ln(2*i-1)\n",
    "    + sp.ln(2*(n-i)+3) - sp.ln(2*(n-i)+1)\n",
    ")\n",
    "der2_f_byhand = (\n",
    "    ((n+1) * (n+1-2*i) * (4*n**2 - 8*(n-i)*(i-1) + 3))\n",
    "    / (i**2 * (2*i-1) * (2*i+1) * (2*n+3-2*i) * (2*n+1-2*i) * (n+1-i)**2)\n",
    ")\n",
    "\n",
    "display(der_f_byhand)\n",
    "sp.plotting.plot(der_f_byhand.subs(n,1000), (i,1,1000))\n",
    "\n",
    "display(der2_f_byhand)\n",
    "sp.plotting.plot(der2_f_byhand.subs(n,1000), (i,1,1000))\n",
    "\n",
    "assert sp.simplify(der_f_comp - der_f_byhand) == 0\n",
    "assert sp.simplify(der2_f_comp - der2_f_byhand) == 0\n",
    "print(\"First and second derivatives are as derived above\")\n",
    "\n",
    "der_f = der_f_byhand\n",
    "der2_f = der2_f_byhand\n",
    "del der_f_byhand, der2_f_byhand\n",
    "del der_f_comp, der2_f_comp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Proof by Monotonicity\n",
    "\n",
    "- $f_n\\left(\\frac{n+1}{2}\\right)\n",
    "    = \\left[xlnx\\left(\\frac{n}{2}+1\\right)\n",
    "    + xlnx\\left(\\frac{n}{2}\\right)\n",
    "    - xlnx\\left(\\frac{n}{2}\\right)\n",
    "    - xlnx\\left(\\frac{n}{2}+1\\right)\\right]\n",
    "    - \\ln\\left(\\frac{n+1}{2}\\right)\n",
    "    + \\ln\\left(\\frac{n+1}{2}\\right)$  \n",
    "    $= 0$\n",
    "- $\\frac{d}{di} f_n\\left(\\frac{n+1}{2}\\right)\n",
    "    = \\ln{(n)} - \\ln{(n+2)} + \\ln{(n)} - \\ln{(n + 2)}\n",
    "    + \\frac{n+1}{\\left(\\frac{n+1}{2}\\right)\\left(\\frac{n+1}{2}\\right)}$          \n",
    "    $= 2\\left(\\frac{2}{n+1} - (\\ln{(n+2)} - \\ln{(n)})\\right)$\n",
    "    - Let $g$ be a function s.t. $\\frac{d^2}{dx^2} g(x) > 0$ for $x\\in[a,b]$\n",
    "        - $\\rightarrow \\int_a^b g(x)dx > (b-a) \\cdot g(b-a)$\n",
    "        - for $g(x) = \\frac1x$, $\\frac{d^2}{dx^2} g(x) = \\frac{2}{x^3} > 0$ for all $x > 0$\n",
    "        - $\\Rightarrow \\int_{n}^{n+2} \\frac{dx}{x} = \\ln{(n+2)} - \\ln{(n)} > \\frac{2}{n+1}$ for $n>0$\n",
    "    - $\\frac{d}{di} f_n\\left(\\frac{n+1}{2}\\right) < 0$\n",
    "- for $i\\in\\left[1, \\frac{n+1}{2}\\right]$, $\\frac{d^2}{dx^2} f_n(i) \\geq 0$\n",
    "- $\\Rightarrow f_n(i) > 0 \\forall i\\in\\left[1, \\frac{n+1}{2}\\right]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f_n((n+1)/2) = 0\n"
     ]
    }
   ],
   "source": [
    "# Show that f at midpoint is zero\n",
    "assert f.subs(i, (n+1)/2).simplify() == 0\n",
    "print(\"f_n((n+1)/2) = 0\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff51244320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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gOxzqzZmSk4HVq9WngBMRaYU/7m/TIQL70CF1JkhWlnpPEN5Rj4i0xh9PvNH93frq6tQbNT3+uPowASLerY80rPPeXlVEfT7i1VcDy5bx7nqkYmCThnXe26suWQLY7eoDcBnWRNTR6baH/emnwNSpwLZtwODBoWwJaQ172KRhPnUtdfn13LFj6lzr/HyGNRF1HroL7DNngLvvVr9gnDAh1K3pvNxuN2w2GywWCzIzM1FTU9Nmufz8fJjNZlgsFqxcudKzvqysDImJiTCbzZg7d+4l6121ahWSkpKQlJSEW265BV9++WWrfRF1dLoL7P/6L+C664D/+Z9Qt6Rzy8vLw/jx47F//35kZGQgNze3VRm3242FCxdix44dKCkpwYIFCzwBPGvWLCxbtgx2ux12ux2FhYUXrTcmJgabN2/GF198gWeffRYzZswI3sESacWlHqt+kVfQ/eMfIkOGiAThafV0CRaLRZxOp4iIHD9+XCwWS6syq1evlpkzZ3qWZ86cKWvWrJHjx4/LsGHD2ix3OfW63W4xmUzttk09rYk0yZfM1U8P+8ABdVbIO+/wknMtqKioQFRUFADAYDCgoqKiVRmHw4Ho6GjPstFohMPhgMPhgMlk8qw3mUxwOBwAAJfLdcl6X3/9dUycONGvx0OkB7qZ1vfss+o9rZOSQt2SzsNqtcLlcnmWRQSKomDRokWtyvrjKq62XFjvpk2bsHz5cnz22WcB2R+RlukisHftUh/p9be/hbolnUtRUVG7v4uKivL0hp1OJyIjI1uVMRqNKC4u9iwfPXoU6enpMBqNKC8vb7He2HxbRYPB0G69e/bswaOPPoqCggL07t37om2fP3++531aWhrS0tIucbREOuDDeErQTJgg8pe/BHOPdCnz5s2TvLw8ERHJy8uTnJycVmWqqqokJiZGqqurPe/dzV9AjBkzRkpKSqSpqUkmTpwoGzduvGi9hw8fltjYWNm6desl2waOYZN2+TSGrfnA3rRJZPBgkdOng7VHuhyVlZUybtw4MZvNYrVaPUFcWloqM2bM8JRbvny5xMbGSlxcnOTn53vWl5aWSnx8vMTGxsrs2bMvWe8jjzwi1113naSkpEhycrKkpqa22zYGNmmYT4Gt6SsdRYCbblJv7PSrXwV6b9RR8EpH0rCOe6Xje+8Bp06pN3giIursNPulY2Mj8N//DfzpT7y/NRERoOEe9t//DvTpA3C6LRGRSpNj2PX1gMWiPvX85psDtRfqqDiGTRrW8cawX3tNvUCGYU1E9DPN9bBPnADi4oCPPgISEgKxB+ro2MMmDetYPez//V/1tqkMayKiljQ1S6S+Hti8Gfi//wt1S4iItEdTPewPPlCn8MXGhrolRETao6nAXrWKVzQSEbVHM186VlcDAwcChw/zftfkG37pSBrWMb50/Oc/gfHjGdZERO3RTGD/4x/A9OmhbgURkXZpYkjk2DEgPl792aOHv2qlzopDIqRh+h8SWbMGuOMOhjUR0cVoIrA5O4SI6NJCHtj796tDIXzkHhHRxYU8sFetAu69F+jaNdQtISLStpBemi6iBvbq1aFsBRGRPoS0h71jB6AowA03hLIVRET6ENLAPvdlo+LTRBcios4hZPOwGxqA6Gj17nxxcb7URNQS52GThulzHvamTWpgM6yJiC5PyAKbl6ITEV2ZkAyJ/PQT0L8/8PXXQL9+3tZC1DYOiZCG6W9I5NNPgcmTGdZERFciJIH9738DAwaEYs9ERPoVksDesgW46aZQ7JmISL+CHthnzwKlpcCNNwZ7z+RPbrcbNpsNFosFmZmZqKmpabNcfn4+zGYzLBYLVq5c6VlfVlaGxMREmM1mzJ0797Lr3bFjB8LCwvDPf/4zMAdGpGFBD+zdu4HBg/lkGb3Ly8vD+PHjsX//fmRkZCA3N7dVGbfbjYULF2LHjh0oKSnBggULPAE8a9YsLFu2DHa7HXa7HYWFhZest6mpCU8//TQyMzODc5BEGhP0wP78c+Dmm4O9V/K3DRs2IDs7GwCQnZ2N9evXtypTWFgIm82G8PBwREREwGazoaCgAE6nE7W1tUhNTQUAZGVleba/WL2vvPIKpk6disjIyEAfHpEmBT2wOX7dMVRUVCAqKgoAYDAYUFFR0aqMw+FAdHS0Z9loNMLhcMDhcMBkMnnWm0wmOBwOAIDL5WqzXofDgfXr12PWrFmcskedVlDv1ieiBnYb/3omDbJarXC5XJ5lEYGiKFi0aFGrskqAbghzrt4nnngCixcvbtEWos4mqIF95AjQ2AjExARzr+StoqKidn8XFRXl6Q07nc42hymMRiOKi4s9y0ePHkV6ejqMRiPKy8tbrDcajQDUXnVb9ZaWlmLatGkQEfz444/YuHEjwsLCMGnSpDbbN3/+fM/7tLQ0pPEJGdQRiIi3ryu2apXInXd6syVpzbx58yQvL09ERPLy8iQnJ6dVmaqqKomJiZHq6mrPe7fbLSIiY8aMkZKSEmlqapKJEyfKxo0bL7veX//61/LOO++02zb1tCbSJF8yN7hj2Fu28AvHjiInJwdFRUWwWCz4+OOP8fTTTwMAdu7ciUcffRQA0Lt3b/z+97/HqFGjMGbMGPzxj39ERPP0oKVLl+Lhhx+G2WxGXFwcJkyYcNF6zxeo4RcirQvqvURGjgSWLgXGjvV2l0SXxnuJkIb51NsIWmDX1qr3DqmsBLp393aXRJfGwCYN08fNn7ZvB5KTGdZERN4KWmBz/JqIyDdBC+zPP+cFM0REvgjKGHZjI9CnD/Dtt8D113u7O6LLwzFs0jDtj2F//TUQGcmwJiLyRVACm+PXRES+C0pgc/yaiMh37GETEelEwAPb6QTcbmDo0EDviYioYwt4YH/+uXopepeQPD2SiKjjCEpgcziEiMh3AQ9sPmGGiMg/AnrhTH29esHMDz8AV1/t7W6IrgwvnCEN0+6FM6WlwPDhDGsiIn8IaGDv3QtkZARyD0REnUdAA/urr9R7YBMRke8CGtjffgvExQVyD0REnQcDm4hIJwI2S+TMGaBXL/XRYGFh3u6C6MpxlghpmDZniRw8CJhMDGsiIn8JWGBzOISIyL8CFth2OwObiMif2MMmItIJBjYRkU4ENLDN5kDVTkTU+QRkWl99PRARAdTVAd26eVs9kXc4rY80THvT+r7/Hhg0iGFNRORPAQlsjl8TEfkfA5u84na7YbPZYLFYkJmZiZqamjbL5efnw2w2w2KxYOXKlZ71ZWVlSExMhNlsxty5cy+r3uLiYqSkpCA+Ph7p6emBOzgijWJgk1fy8vIwfvx47N+/HxkZGcjNzW1Vxu12Y+HChdixYwdKSkqwYMECTwDPmjULy5Ytg91uh91uR2Fh4UXrrampwWOPPYb3338fe/fuxdq1a4N3sEQawcAmr2zYsAHZ2dkAgOzsbKxfv75VmcLCQthsNoSHhyMiIgI2mw0FBQVwOp2ora1FamoqACArK8uzfXv1rlq1ClOmTIHRaAQA9O3bN+DHSKQ1DGzySkVFBaKiogAABoMBFRUVrco4HA5ER0d7lo1GIxwOBxwOB0wmk2e9yWSCw+EAALhcrjbrtdvtqKqqQnp6OlJTU/H3v/89YMdGpFV+n8dx6hRQWQmc998p6ZTVaoXL5fIsiwgURcGiRYtalVUUn2YrtetcvQ0NDSgrK8Mnn3yCkydPYuzYsRg7dixiY2Pb3G7+/Pme92lpaUhLSwtI+4iCye+B/d13QEwM0CXgz2OnQCsqKmr3d1FRUZ7esNPpRGRkZKsyRqMRxcXFnuWjR48iPT0dRqMR5eXlLdafG+owGAxt1msymdC3b1/06NEDPXr0wC9+8Qt88cUXlxXYRB2F32OVwyGdw6RJk7BixQoA6kyQyZMntyqTmZmJoqIi1NTUwO12o6ioCJmZmTAYDAgPD8f27dshIli5cqVn+/bqnTx5Mj777DM0Njbi1KlTKCkpwbBhw4JyrESaISLevtr0/PMiTz3V3m+po6isrJRx48aJ2WwWq9UqbrdbRERKS0tlxowZnnLLly+X2NhYiYuLk/z8fM/60tJSiY+Pl9jYWJk9e/Yl6xUR+dOf/iTDhw+XhIQEefnll9ttm3paE2mSL5nr/0vTH3oIuPFG4NFHvf5/CJFPeGk6aZi2Lk3nTZ+IiAKDY9hERDrh18A+cUJ96G7//v6slYiIAD8H9nffAbGxQICm5BIRdWp+DWwOhxARBQ4Dm4hIJxjYREQ6wcAmItIJBjYRkU74LbCrq9WH7zbfGZOIiPzMb4F9rnfNKX1ERIHht8C22zkcQkQUSH7vYRMRUWD4NbB50yciosBhD5uISCf8FthNTep9RIiIKDD88gCDU6eA664DfvqJs0Qo9PgAA9Kw0D/AwOVS518zrImIAscvge10AgaDP2oiIqL2+LWHTUREgcPAJiLSCQ6JEBHpBHvYREQ6wcAmItIJDokQEekEe9hERDrBwCYi0gmfA/vkSaChAejVyx/NISKi9vgc2LwsvXNyu92w2WywWCzIzMxETU1Nm+Xy8/NhNpthsViwcuVKz/qysjIkJibCbDZj7ty5l6z3xIkTmDRpEpKTk5GQkIAVK1YE9PiItMhvgU2dS15eHsaPH4/9+/cjIyMDubm5rcq43W4sXLgQO3bsQElJCRYsWOAJ4FmzZmHZsmWw2+2w2+0oLCy8aL1Lly7FiBEjsHv3bmzatAlPPvkkGhoagnfARBrgc2BzhkjntGHDBmRnZwMAsrOzsX79+lZlCgsLYbPZEB4ejoiICNhsNhQUFMDpdKK2thapqakAgKysLM/27dWrKApqa2sBALW1tejTpw+6desW8OMk0hKfz3j2sDuniooKRDV/8AaDARUVFa3KOBwOREdHe5aNRiMcDgccDgdMJpNnvclkgsPhAAC4XK4263388ccxadIk9O/fH3V1dXjrrbcCdmxEWsXApnZZrVa4XC7PsohAURQsWrSoVVklQF9inKu3oKAAKSkp+OSTT/D999/DarViz5496NmzZ5vbzZ8/3/M+LS0NaWlpAWkfUTD5HNhOJxAf74+mkNYUFRW1+7uoqChPb9jpdCIyMrJVGaPRiOLiYs/y0aNHkZ6eDqPRiPLy8hbrjUYjALVX3Va9K1aswDPPPAMAGDJkCAYPHoxvvvkGo0aNarN95wc2UUfBLx3JK5MmTfLM1MjPz8fkyZNblcnMzERRURFqamrgdrtRVFSEzMxMGAwGhIeHY/v27RARrFy50rP9+fWuWLHCs37AgAH46KOPAKjDJna7HTExMYE/UCItERFvXyIictNNIps3C3UylZWVMm7cODGbzWK1WsXtdouISGlpqcyYMcNTbvny5RIbGytxcXGSn5/vWV9aWirx8fESGxsrs2fPvmS9x44dE5vNJgkJCZKQkCCrVq1qt23qaU2kSb5kru/PdBwyBCgo4BPTSTv4TEfSsNA+05FDIkREweFTYNfVAY2NwLXX+qs5RETUHp8C2+VSL5rhZelERIHnc2BzOISIKDgY2EREOuFTYPM+IkREwcMeNhGRTjCwiYh0gkMiREQ6wR42EZFOMLCJiHSCQyJERDrhdWDX1QEiQDv3jyciIj/zOrD5tHQiouDyOrA5HEJEFFw+97CJiCg4GNhERDrBIREiIp1gD5uISCcY2EREOsEhESIinWAPm4hIJxjYREQ64XVg87J0IqLg8jqw+bR0IqLg8jqwORxCRBRcDGwiIp3waUiEiIiChz1s8orb7YbNZoPFYkFmZiZqamraLJefnw+z2QyLxYKVK1d61peVlSExMRFmsxlz5871rF+3bh3i4+PRtWtXlJWVtagrNzcXcXFxGDZsGD788MPAHBiRhjGwySt5eXkYP3489u/fj4yMDOTm5rYq43a7sXDhQuzYsQMlJSVYsGCBJ9hnzZqFZcuWwW63w263o7CwEACQkJCAd999F7/85S9b1LVv3z68/fbb2LdvHzZu3Ijf/va3EJHAHyiRhnBIhLyyYcMGZGdnAwCys7Oxfv36VmUKCwths9mOgG/BAAAFEklEQVQQHh6OiIgI2Gw2FBQUwOl0ora2FqmpqQCArKwsz/YWiwVxcXGtwnjDhg2YNm0aunXrhkGDBiEuLg7bt28P8FESaQt72OSViooKRDWfBAaDARUVFa3KOBwOREdHe5aNRiMcDgccDgdMJpNnvclkgsPhuOj+2quLqDPp5u2GDOyOz2q1wuVyeZZFBIqiYNGiRa3KKpyUTxRwDGxqV1FRUbu/i4qKgsvlQlRUFJxOJyIjI1uVMRqNKC4u9iwfPXoU6enpMBqNKC8vb7HeaDRetC1Xus38+fM979PS0pCWlnbR+ol0QUS8fVEnNm/ePMnLyxMRkby8PMnJyWlVpqqqSmJiYqS6utrz3u12i4jImDFjpKSkRJqammTixImycePGFtumpaVJaWmpZ/mrr76S5ORkOX36tBw4cECGDBkiTU1NbbZNPa2JNMmXzGVgk3cqKytl3LhxYjabxWq1eoK4tLRUZsyY4Sm3fPlyiY2Nlbi4OMnPz/esLy0tlfj4eImNjZXZs2d71r/77rtiMpmkR48eYjAYZMKECZ7fPf/88zJkyBAZOnSoFBYWtts2BjZpmE+BrYj3U6M4p4o0SVEUTvkjrfLpyx6vZ4kQEVFwMbCJiHSCgU1EpBMMbCIinWBgExHpBAObiEgnGNhERDrBwCYi0gkGNhGRTjCwiYh0goFNRKQTDGwiIp1gYBMR6QQDm4hIJxjYREQ6wcAmItIJBjYRkU4wsImIdIKBTUSkEwxsIiKdYGATEekEA5uISCcY2EREOsHAJiLSCQY2EZFOMLCJiHSCgU1EpBMMbCIinWBgk1fcbjdsNhssFgsyMzNRU1PTZrn8/HyYzWZYLBasXLnSs76srAyJiYkwm82YO3euZ/26desQHx+Prl27oqyszLP+o48+wqhRo5CUlITU1FRs2rQpcAdHpFUi4u2LOrF58+bJ4sWLRUQkLy9PcnJyWpWpqqqSmJgYqa6uFrfb7XkvIjJ69GjZvn27iIhMnDhRCgoKRETkm2++EbvdLunp6bJz505PXbt375bjx4+LiMjevXvFaDS22zb1tCbSJF8ylz1s8s6GDRuQnZ0NAMjOzsb69etblSksLITNZkN4eDgiIiJgs9lQUFAAp9OJ2tpapKamAgCysrI821ssFsTFxUFEWtSVlJQEg8EAABgxYgTq6+tx9uzZQB4ikeYwsMkrFRUViIqKAgAYDAZUVFS0KuNwOBAdHe1ZNhqNcDgccDgcMJlMnvUmkwkOh+Oy971u3TqMHDkSYWFhPhwBkf50C3UDSLusVitcLpdnWUSgKAoWLVrUqqyiKEFp01dffYVnnnkGRUVFQdkfkZYoF/7Tk+hyKIqyD0CaiLgURTEA2CQiwy4oM625zMzm5dcAbALw7/PLN5f7pYjMOm/bTQCeFJGy89aZAHwMIFtEtl2kbQJgwXmrikWk2KcDJtIA9rDJW+8B+DWAxQCyAWxoo0whgP+nKEo41OE3K4CnRaRaUZQaRVFGA9gBIAvAy21s7+m2N9fxPoCci4U1AIhIcLr7REHGMWzy1mIAVkVR9gMYByAPABRFuUFRlL8BgIi4ATwHoBRACYAFIlLdvP1jAJYBsAP4VkQKmre/Q1GUcgA3AnhfUZSNzeUfBzAEwB8URdmlKEqZoih9g3GgRFrBIREiIp1gD5uISCcY2EREOsHAJiLSCQY2EZFOMLCJiHSCgU1EpBMMbCIinWBgExHpxP8H78xJObe+IN0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff512c0da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff53548ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Show that derivative at midpoint is less than zero\n",
    "for i_first, i_last in zip((1,10,100), (10,100,1000)):\n",
    "    sp.plotting.plot(der_f.subs(i, (n+1)/2), (n, i_first, i_last))#.cancel().collect(sp.ln(n)).collect(sp.ln(n+2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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73/kOAMePH2fJkiX09PTgnGPx4sX09PQwY8aMEa9jZuR6/4oUkY2lsypmybvu7m5CoRC1tbWUlJQQj8dpb28f1uaqq67i61//OmbD37/bt2+noaGBGTNmUFZWRkNDA52dnYUcvkjRKZgl75LJJDU1Ndn16upqksnkJfUNBoMX3Fdkopiaa6eCWXzX0tKSXY7FYsRisaKNRSRfFMySd8FgkP7+/uz6wMAAwQu8KT4YDLJ79+5hfVeuXHne9kODWWSi0JN/knfRaJTe3l76+vpIpVIkEgkaGxvP237oBbzVq1ezY8cOPvroI44fP86OHTtYvXp1IYYt4g3NMUveBQIB2traaGhoYP78+cTjcerq6mhubmbbtm0AvPHGG9TU1PDMM8+wbt06rrnmGgDKy8u55557WLJkCcuWLaO5uZmysrJiHo5IweW8Xe7LX8YdO1bA0YhcBN0uJx7T7XIiIhOJgllExDO6+Cci4pmcwTzkPn8RESmQnMHc11eoYYiIyFk5gzmdLtQwRETkLF38ExHxjIJZRMQzCmYREc8omEVEPKNgFhHxjIJZRMQzCmYREc/oPmYREc+oYhYR8YwqZhERz6hiFhHxzKh/9lPhLCJSWDmD2UzBLCJSaKMGs+aZRUQKK2cwT5miillEpNBGDWZVzCIihaWpDBERz2gqQ8ZFZ2cnkUiEcDhMa2vriP2pVIp4PE4oFGL58uX09/cD0NfXx+WXX059fT319fWsX7++0EMXKbqpuXaqYpZLkU6n2bBhA11dXVRVVRGNRrn55puJRCLZNlu2bKGiooJDhw7x61//mo0bN5JIJAC4+uqr6enpKdbwRYpOFbPkXXd3N6FQiNraWkpKSojH47S3tw9r097ezh133AHArbfeSldXV3af05tOJjnNMUveJZNJampqsuvV1dUkk8nztgkEApSVlfHhhx8C8N5777F48WJWrlzJK6+8UriBi3gi51TGvHlw5kyhhiKT2dkqubKykv7+fsrLy+np6eGWW27hwIEDTJ8+/Zz9WlpassuxWIxYLFaA0YqMr5zB/O67msqQixcMBrMX8wAGBgYIBoPD2lRXV3P06FGqqqo4c+YMJ0+epKKiAoDS0lIA6uvrmTNnDu+++y719fXnfK2hwSwyUeScyggEVDHLxYtGo/T29tLX10cqlSKRSNDY2Diszbe//W22bt0KwNNPP82qVasAOHbsGOnM/Nnhw4fp7e3la1/7WmEPQKTIclbMCma5FIFAgLa2NhoaGkin0zQ1NVFXV0dzczPRaJS1a9fS1NTE7bffTigUYubMmdk7Mvbu3cu9995LaWkpU6ZM4ZFHHqGsrKzIRyRSWJbrCngwiHv9daiuLuCIRC6QmekODvGVjaVzzqmMqVNVMYuIFNqoc8x//WuhhiIiIqCLfyIi3lEwi4h4RsEsIuIZBbOIiGcUzCIinlEwi4h4RsEsIuIZBbOIiGcUzCIinskZzBUVevJPRKTQcgbzqVODPyIiUjg5g/kLX4BUqlBDERERGCWYS0tVMYuIFNqoFbOCWUSksEatmDWVISJSWDmDubpaFbOISKGNOpUxMFCooYiICIwSzF/9KnzwQaGGIiIicAHB/PvfF2ooIiICowRzZaWCWUSk0FQxi4h4JmcwX3EFHDumP2QEsHv37mIPwRsXci46OzuJRCKEw2FaW1tH7E+lUsTjcUKhEMuXL6e/vz+7b9OmTYRCIerq6njxxRfzOfS80/viMzoXnzGz2Fj65wzmkhIoK9MFQNCbbqjRzkU6nWbDhg1s376dt99+myeffJJ33nlnWJstW7ZQUVHBoUOH+NGPfsTGjRsBOHDgAE899RQHDx7khRdeYP369TjnxutQxkzvi8/oXAwTG0vnnMEM8P3vww9/CKdPj+VlZDLp7u4mFApRW1tLSUkJ8Xic9vb2YW3a29u54447ALj11lvZtWsXAB0dHcTjcaZOncqsWbMIhUJ0d3cX/BhEimnqaA3uuw/uvBPmz4dp0wYr6MpK+OgjMBv8AaithaNHh/edPh0+/XTkv/noozBz5tgHL35KJpPU1NRk16urq0eE69A2gUCAGTNm8OGHH5JMJlm+fHm2XTAYJJlMFmbgIp6wXF8Tzczf75AiIh5zztml9s0ZzCKXwsyuBVqcczdl1n8KOOdc65A2L2TavG5mAeAD59yVn29rZp1As3Pu9cIfiUhxjDrHLHIJfgNcbWa1ZlYKxIGOz7V5Hrgjs3wbsCuz3AHEzazUzGYDVwOaZJZJZdQ5ZpGL5Zw7Y2YbgBcZ/OW/xTl30Mz+G/Ab59w2YAvwuJkdAv4fg+GNc+6AmT0FHABOA+udvtbJJKOpDBERz2gqAzCzLWb2BzP77ZBt5Wb2opn9zsy2m9mMIfv+u5kdMrM3zWxhcUY9Psys2sx2mdnbZvaWmf3nzPZJdz7M7Atm9rqZ7c+ci+bM9llm9s9m9q6ZPWlmUzPbS80skTkXr5nZVcU9gvwzsylm1mNmHZn1SXkuzOw9M/uXzHujO7Mtb58RBfOg/wms/ty2nwI7nXNzGZz/vBvAzNYAc5xzIeAuYHMhB1oAfwV+7JybDywH/sHMIkzC8+GcOwWsdM4tAhYCa8xsGdAKPOicCwMngKZMlybgw8y5+Efg/iIMe7z9kMFpprMm67lIAzHn3CLn3NLMtvx9Rpxz+hmczqkFfjtk/R3gK5nlrwIHM8ubge8NaXfwbLuJ+AM8B3xrsp8P4HLgDWAp8K/AlMz2a4EXMsudwLLMcgD4Y7HHnedzUA3sYPCpto7Mtj9O0nNxBJj5uW15+4yoYj6/K51zfwBwzv0e+EpmexAY+ihNMrNtwjGzWQxWiv/M4Btp0p2PzFf3/cDvGQyl/wuccM6lM00G+Ox4s+fCOXcGOGFmFQUe8nh6CPgJ4ADMbCZwfJKeCwdsN7PfmNl/zGzL22dEd2VcuEl1ldTMpgPPAD90zn1yjoeNJsX5yITOIjP7EvB/gMhFdL/kBwx8Y2Z/C/zBOffm5/5Az4Ue44Q5FxnfdM59YGZXAC+a2e8Y+Zm45M+IKubz+4OZfQXAzL7K4NdXGPxtVzOkXXVm24SRuYDzDPC4c+7sH7mYtOcDwDl3EtjN4Lx7mZmd/ewMPd7sucg8NPMl59yHBR7qePkm0Ghmh4EngVXAPwEzJuG5wDn3Qea/f2Rwum8pefyMKJg/Ywz/rd4B3JlZvhNoH7L930P2CbcTZ7++TCD/AzjgnPunIdsm3fkwsy+fvbJuZpcBNzJ44eslBh+KgcGHZIaei3M9NPNvnnPuZ865q5xzX2PwnvNdzrm/YxKeCzO7PPONEjP7ItAAvEU+PyPFnkT34Qf4X8D7wCmgH/g+UA7sBH7H4IMSZUPatwG9wL8A9cUef57PxTeBM8CbwH6gB7gJqJhs5wO4JnP8bwK/Bf5rZvts4HXgXeDXQElm+xeAp4BDDM7Lzyr2MYzTeVnBZxf/Jt25yBzz2c/HW8BPM9vz9hnRAyYiIp7RVIaIiGcUzCIinlEwi4h4RsEsIuIZBbOIiGcUzCIinlEwi4jkmZm9Mqb+uo9ZRMQvqphFRPLMzD4eS38Fs4hI/o1pKkLBLCLiGQWziIhnFMwiIvk3pv8xgIJZRCT/xjTHrNvlREQ8o4pZRMQzCmYREc8omEVEPKNgFhHxjIJZRMQzCmYREc8omEVEPPP/AZVVK2KjhymtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50ffb940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50f416d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50f62a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff50f1d668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Show that second derivative is positive from 1 to midpoint\n",
    "for i_first in (1,10,100,400):\n",
    "    sp.plotting.plot(der2_f.subs(n, n_val), (i, i_first, n_val/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### QED"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Second: Prove that $nCk = \\Omega\\left(n\\hat{C}k\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Assessment at Midpoint\n",
    "- $\\frac{n\\hat{C}k}{nCk} = \\exp{\\left(\\sum_{i=1}^{k} f_n(i)\\right)}$\n",
    "- $f_n(i) \\geq 0$ for $i\\in\\left[1,\\frac{n}{2}\\right]$\n",
    "- $\\Rightarrow \\max\\left(\\left.\\frac{n\\hat{C}k}{nCk}\\right| k\\in[0,n]\\right)\n",
    "    = \\frac{n\\hat{C}\\left(\\frac{n}{2}\\right)}{nC\\left(\\frac{n}{2}\\right)}$\n",
    "- ETS that $\\sup\\left(\\frac{n\\hat{C}\\left(\\frac{n}{2}\\right)}{nC\\left(\\frac{n}{2}\\right)}\\right) < \\infty$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $\\frac{n\\hat{C}\\left(\\frac{n}{2}\\right)}{nC\\left(\\frac{n}{2}\\right)}\n",
    "    = \\exp\\left(\n",
    "        \\int_{\\frac12}^{\\frac{n+1}{2}} \\left[\\ln{(n+1-x)}-\\ln{(x)}\\right]dx\n",
    "        - \\sum_{i=1}^{\\frac{n}{2}} \\left[\\ln{(n+1-i)} - \\ln{(i)}\\right]\n",
    "        \\right)$  \n",
    "    $= \\exp\\sum_{i=1}^{\\frac{n}{2}} \\left(\n",
    "        \\left(\\ln{(i)}-\\int_{i-\\frac12}^{i+\\frac12} \\ln{(x)}dx\\right)\n",
    "        - \\left(\\ln{(n+1-i)}-\\int_{i-\\frac12}^{i+\\frac12} \\ln{(n+1-x)}dx\\right)\n",
    "        \\right)$ \n",
    "\n",
    "[Euler-Maclaurin formula](https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\log{\\left (\\left(n + 1\\right)^{- n - 1} \\left(2 n + 1\\right)^{n + \\frac{1}{2}} \\right )}}{\\sum_{i=1}^{\\frac{n}{2}} \\left(- \\log{\\left (i \\right )} + \\log{\\left (- i + n + 1 \\right )}\\right)}$$"
      ],
      "text/plain": [
       "   ⎛       -n - 1          n + 1/2⎞\n",
       "log⎝(n + 1)      ⋅(2⋅n + 1)       ⎠\n",
       "───────────────────────────────────\n",
       "   n                               \n",
       "   ─                               \n",
       "   2                               \n",
       "  ___                              \n",
       "  ╲                                \n",
       "   ╲   (-log(i) + log(-i + n + 1)) \n",
       "   ╱                               \n",
       "  ╱                                \n",
       "  ‾‾‾                              \n",
       " i = 1                             "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, args)\u001b[0m\n\u001b[1;32m    193\u001b[0m             \u001b[0;31m#The result can be sympy.Float. Hence wrap it with complex type.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 194\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcomplex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambda_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    195\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimag\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1e-7\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    315\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__call__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 316\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambda_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    317\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x0)\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__complex__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    243\u001b[0m         \u001b[0mre\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mim\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_real_imag\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 244\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mcomplex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mre\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mim\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    245\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__float__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    238\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"can't convert complex to float\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 239\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"can't convert expression to float\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    240\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: can't convert expression to float",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/evalf.py\u001b[0m in \u001b[0;36mevalf_sum\u001b[0;34m(expr, prec, options)\u001b[0m\n\u001b[1;32m   1170\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mb\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mInfinity\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1171\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1172\u001b[0m         \u001b[0;31m# Use fast hypergeometric summation if possible\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNotImplementedError\u001b[0m: ",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-58-5a62ab1dabfc>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      9\u001b[0m \u001b[0mdisplay\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnCk_est_ln_mid\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mnCk_ln_mid\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     10\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 11\u001b[0;31m \u001b[0msp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplotting\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnCk_est_ln_mid\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mnCk_ln_mid\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m10\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mplot\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m   1293\u001b[0m     \u001b[0mplots\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPlot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1294\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1295\u001b[0;31m         \u001b[0mplots\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1296\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mplots\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1297\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mshow\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    194\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    195\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbackend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 196\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_backend\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    197\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    198\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msave\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpath\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mshow\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1027\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1028\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1029\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mprocess_series\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1030\u001b[0m         \u001b[0;31m#TODO after fixing https://github.com/ipython/ipython/issues/1255\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1031\u001b[0m         \u001b[0;31m# you can uncomment the next line and remove the pyplot.show() call\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/plot.py\u001b[0m in \u001b[0;36mprocess_series\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    906\u001b[0m             \u001b[0;31m# Create the collections\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    907\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_2Dline\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 908\u001b[0;31m                 \u001b[0mcollection\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mLineCollection\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_segments\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    909\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0madd_collection\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcollection\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    910\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_contour\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, args)\u001b[0m\n\u001b[1;32m    229\u001b[0m                                                     \u001b[0muse_evalf\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    230\u001b[0m                                                     complex_wrap_evalf=True)\n\u001b[0;32m--> 231\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambda_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    232\u001b[0m                 warnings.warn('The evaluation of the expression is'\n\u001b[1;32m    233\u001b[0m                         \u001b[0;34m' problematic. We are trying a failback method'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/plotting/experimental_lambdify.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    314\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    315\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__call__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 316\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambda_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    317\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    318\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x0)\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/evalf.py\u001b[0m in \u001b[0;36mevalf\u001b[0;34m(self, n, subs, maxn, chop, strict, quad, verbose)\u001b[0m\n\u001b[1;32m   1392\u001b[0m             \u001b[0moptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'quad'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mquad\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1393\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1394\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprec\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1395\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1396\u001b[0m             \u001b[0;31m# Fall back to the ordinary evalf\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/evalf.py\u001b[0m in \u001b[0;36mevalf\u001b[0;34m(x, prec, options)\u001b[0m\n\u001b[1;32m   1284\u001b[0m     \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1285\u001b[0m         \u001b[0mrf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevalf_table\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1286\u001b[0;31m         \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprec\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1287\u001b[0m     \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1288\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/evalf.py\u001b[0m in \u001b[0;36mevalf_sum\u001b[0;34m(expr, prec, options)\u001b[0m\n\u001b[1;32m   1182\u001b[0m             \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mi\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mprec\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1183\u001b[0m             s, err = expr.euler_maclaurin(m=m, n=n, eps=eps,\n\u001b[0;32m-> 1184\u001b[0;31m                 eval_integral=False)\n\u001b[0m\u001b[1;32m   1185\u001b[0m             \u001b[0merr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0merr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1186\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0merr\u001b[0m \u001b[0;34m<=\u001b[0m \u001b[0meps\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/concrete/summations.py\u001b[0m in \u001b[0;36meuler_maclaurin\u001b[0;34m(self, m, n, eps, eval_integral)\u001b[0m\n\u001b[1;32m    633\u001b[0m                 \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    634\u001b[0m                     \u001b[0mterm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 635\u001b[0;31m                     \u001b[0;32mif\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mterm\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0meps\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mterm\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    636\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mterm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    637\u001b[0m                     \u001b[0ms\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mterm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__abs__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    113\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__abs__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    114\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mAbs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 115\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mAbs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0m_sympifyit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'other'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mNotImplemented\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    425\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    426\u001b[0m         \u001b[0mevaluate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'evaluate'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_evaluate\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 427\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    428\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    429\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    248\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 250\u001b[0;31m             \u001b[0mevaluated\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    251\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mevaluated\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mevaluated\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/functions/elementary/complexes.py\u001b[0m in \u001b[0;36meval\u001b[0;34m(cls, arg)\u001b[0m\n\u001b[1;32m    452\u001b[0m             \u001b[0munk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    453\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mt\u001b[0m \u001b[0;32min\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 454\u001b[0;31m                 \u001b[0mtnew\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    455\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mtnew\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    456\u001b[0m                     \u001b[0munk\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtnew\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    425\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    426\u001b[0m         \u001b[0mevaluate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'evaluate'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_evaluate\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 427\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    428\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    429\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    248\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 250\u001b[0;31m             \u001b[0mevaluated\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    251\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mevaluated\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mevaluated\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/functions/elementary/complexes.py\u001b[0m in \u001b[0;36meval\u001b[0;34m(cls, arg)\u001b[0m\n\u001b[1;32m    447\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Bad argument type for Abs(): %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    448\u001b[0m         \u001b[0;31m# handle what we can\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 449\u001b[0;31m         \u001b[0marg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msignsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    450\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    451\u001b[0m             \u001b[0mknown\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36msignsimp\u001b[0;34m(expr, evaluate)\u001b[0m\n\u001b[1;32m    373\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    374\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 375\u001b[0;31m     \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msub_post\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msub_pre\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    376\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    377\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/simplify/cse_opts.py\u001b[0m in \u001b[0;36msub_pre\u001b[0;34m(e)\u001b[0m\n\u001b[1;32m     24\u001b[0m         \u001b[0mnegs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     25\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msorted\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0matoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mAdd\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdefault_sort_key\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 26\u001b[0;31m             \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mreps\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcould_extract_minus_sign\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     27\u001b[0m                 \u001b[0mnegs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mNegativeOne\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     28\u001b[0m         \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnegs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mcould_extract_minus_sign\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   2198\u001b[0m                 \u001b[0;31m# We choose the one with less arguments with minus signs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2199\u001b[0m                 \u001b[0mall_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2200\u001b[0;31m                 \u001b[0mnegative_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcould_extract_minus_sign\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2201\u001b[0m                 \u001b[0mpositive_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mall_args\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mnegative_args\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2202\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mpositive_args\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0mnegative_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m   2198\u001b[0m                 \u001b[0;31m# We choose the one with less arguments with minus signs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2199\u001b[0m                 \u001b[0mall_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2200\u001b[0;31m                 \u001b[0mnegative_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcould_extract_minus_sign\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2201\u001b[0m                 \u001b[0mpositive_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mall_args\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mnegative_args\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2202\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mpositive_args\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0mnegative_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mcould_extract_minus_sign\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   2191\u001b[0m         \u001b[0mself_has_minus\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mextract_multiplicatively\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2192\u001b[0m         negative_self_has_minus = (\n\u001b[0;32m-> 2193\u001b[0;31m             (negative_self).extract_multiplicatively(-1) is not None)\n\u001b[0m\u001b[1;32m   2194\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mself_has_minus\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mnegative_self_has_minus\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2195\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself_has_minus\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mextract_multiplicatively\u001b[0;34m(self, c)\u001b[0m\n\u001b[1;32m   2055\u001b[0m             \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2057\u001b[0;31m                 \u001b[0mnewarg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mextract_multiplicatively\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2058\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mnewarg\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2059\u001b[0m                     \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnewarg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mextract_multiplicatively\u001b[0;34m(self, c)\u001b[0m\n\u001b[1;32m   1985\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mextract_multiplicatively\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1986\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1987\u001b[0;31m         \u001b[0mquotient\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1988\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Number\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1989\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mInfinity\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/decorators.py\u001b[0m in \u001b[0;36m__sympifyit_wrapper\u001b[0;34m(a, b)\u001b[0m\n\u001b[1;32m     89\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'_op_priority'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     90\u001b[0m                     \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mstrict\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 91\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     92\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mSympifyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     93\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mretval\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.5/dist-packages/sympy/core/numbers.py\u001b[0m in \u001b[0;36m__div__\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m   1174\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mNumber\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mother\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m0\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mglobal_evaluate\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1175\u001b[0m             \u001b[0mrhs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprec\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_as_mpf_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_prec\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1176\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mFloat\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_new\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmlib\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmpf_div\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_mpf_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrhs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprec\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrnd\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprec\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1177\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mNumber\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__div__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1178\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7eff5111aac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nCk_mid = nCk.subs(k, n/2).simplify()\n",
    "nCk_est_mid = nCk_est.subs(k, n/2).simplify()\n",
    "#display(nCk_est_mid / nCk_mid)\n",
    "\n",
    "nCk_ln_mid = nCk_ln.subs(k, n/2).simplify()\n",
    "nCk_est_ln_mid = nCk_est_ln.subs(k, n/2).simplify()\n",
    "display(nCk_est_ln_mid / nCk_ln_mid)\n",
    "\n",
    "sp.plotting.plot(nCk_est_ln_mid.subs(n,2*n) / nCk_ln_mid.subs(n,2*n), (n,1,10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comparing Estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "'''def plot_nckest(n_val, ests, omitted=0):\n",
    "    plot = sp.plotting.plot(\n",
    "        sp.binomial(n_val, k), (k, 0, n_val),\n",
    "        line_color=\"k\",\n",
    "        title=\"nCk & Estimates (n = \" + str(n_val) + \")\",\n",
    "        xlabel=\"k-values\",\n",
    "        yscale=\"log\",\n",
    "        axis_center=(0,1),\n",
    "        legend=True,\n",
    "        show=False\n",
    "    )\n",
    "    \n",
    "    for i, func in enumerate(ests[omitted:], omitted):\n",
    "        plot.extend(sp.plotting.plot(\n",
    "            func.subs(n,n_val), (k, 0, n_val),\n",
    "            line_color=est_styles[i],\n",
    "            title=est_labels[i],\n",
    "            show=False\n",
    "        ))\n",
    "    \n",
    "    plot.show()\n",
    "\n",
    "plot_nckest(2, nCk_ests_sp)\n",
    "plot_nckest(3, nCk_ests_sp)\n",
    "plot_nckest(10, nCk_ests_sp)\n",
    "plot_nckest(100, nCk_ests_sp)\n",
    "plot_nckest(100, nCk_ests_sp, omitted=2)\n",
    "plot_nckest(120, nCk_ests_sp, omitted=3)'''\n",
    "pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hyper_sq(x):\n",
    "    return x ** x\n",
    "\n",
    "nCk_ests = (\n",
    "    lambda n,k: n ** k,\n",
    "    lambda n,k: (n-k+1) ** k,\n",
    "    lambda n,k: n ** (k*(n-k)/(n-1)),\n",
    "    lambda n,k: np.sqrt(.5) * hyper_sq(n+.5) / (hyper_sq(n-k+.5) * hyper_sq(k+.5))\n",
    ")\n",
    "\n",
    "est_labels = (\n",
    "    \"$n^k$\",\n",
    "    \"${(n-k+1)}^k$\",\n",
    "    \"$n^{\\\\left(\\\\frac{k(n-k)}{n-1}\\\\right)}$\",\n",
    "    \"$\\\\sqrt{\\\\frac{1}{2}} \\\\cdot \\\\frac{^2 \\\\left(n+\\\\frac{1}{2}\\\\right)}{\"\n",
    "        \"^2 \\\\left(n-k+\\\\frac{1}{2}\\\\right)\"\n",
    "        \"\\\\cdot ^2 \\\\left(k+\\\\frac{1}{2}\\\\right)\"\n",
    "        \"}$\"\n",
    ")\n",
    "\n",
    "est_styles = (\n",
    "    \"m--\",\n",
    "    \"r--\",\n",
    "    \"y--\",\n",
    "    \"g--\",    \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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lE5GvgZkichGw+kQHU0rNAmaBfVitM4NrxqiszGXHjlHk5i4nLOweOnd+vUlNENhU+fkNoF+/OBISbiQx8VaKijYTHf2SbtG1YC73L6+U+rjGn0uAScal0VxBevp75OWtomvXDwgLu93oOC2Ku3swZ521gr17p3Lw4Kv4+p5LcPANRsfSDGJEwUgD2td4H+FYpmnHsNnKMZk8iIx8lMDAq2jVqpfRkVokk8mNrl3fcjwWdhBgv1HSftVYa0mM+BffAHQRkWgRcQdGAd8bkENzYRkZC1i/vjtlZamImHWxcAH+/hcjIhQXJ7BhQy+KirYYHUlrZM4eVjsf+BPoJiKpIjJJKVUF3AssAxKBRUqpBGfm0JqW1NTpJCaOxsOjvX7cqAtSSmG1FrB58yDy8n4zOo7WiPRcUprLUEqxf/+TpKS8RFDQdXTv/tnR6bk111JWdpCtWy+ntDSJ2NjPjw7F1ZomPZeU1uSkpr5BSspLhIbeQY8ei3SxcGGenu3p23cNvr59SUi4kZyc5UZH0hqBy42S0lqu0NDbMZm8CAuboqf4aALc3AI566wVpKT8D3//i42OozUC3cLQDGWzVZCc/CxWazEWix/h4XfpYtGEmM0+REc/g8nkQWXlEQ4dmmN0JM2JdMHQDGOzVZCQcAPJyU+Tk7PU6DjaGUpNncGuXZM4cOAFo6NoTqIvSWmGsBeLmzhy5Ae6dHmHtm2vNzqSdoaiov5NWdl+9u+fBpjp0OGxU+6jNS26YGiNzmarZMeOURw58h1durxFeLieV7I5EDETE/MRSlnZv/9xRMxERv7T6FhaA9KXpLRGV16eSn7+H3Tu/Cbh4fcYHUdrQPai8QnBwaNIT3/v6FT0WvOgWxhao6m+58fLK5oBA3bi5uZvcCLNGUwmCzEx86iqOoLF0sroOFoD0i0MrVEopdi37xGSkv6FUkoXi2bOZLLg7h6CzVbFzp23k5n5pdGRtAagC4bWKA4ceJ7U1Nex2cqMjqI1IqUqKClJJDHxFnJylhkdRztDumBoTpee/gHJyf8mJGQsnTu/qe+zaEHMZm969VqMt3cs27dfT0GBnrqnKdMFQ3Oq7Owf2b17CgEBw+jWbbaeErsFcnPzp3fvn3BzC2LbtispLd1ndCTtNOn/ezWnstlK8PMbSGzsF5hMbkbH0Qzi4RHKWWctw2TyoqzsgNFxtNOkZ6vVnMJmqzr6KE/9sB2tms1WcfTxukopfXnSRejZajXDVFbmsnFjPzIyFgDoYqEdVV0sUlNnsmPHKJSyGZxIqw/9f7LWoOxTflxHSclOPDzCjY6juSilKsjKWsT+/U8aHUWrB33jntZglFLs3j2FvLxVdO/+Kf7+FxkdSXNREREPUVq6l5SUl/Hy6kJo6G1GR9LqwGVaGCLSXUTeE5EvReQux7JYEVkkIu+KyA1GZ9ROLjX1DQ4f/ogOHZ4mJORWo+NoLkxE6Nx5Jm3aXMbu3VPIz19rdCStDpz9TO85IpIpItuPWz5MRHaJyF4ReQxAKZWolJoC3ARc4Nh0ODBTKXUXMM6ZWbUzV1l5hLZtbyAq6t9GR9GaAJPJQmzsAry8OlFUFG90HK0OnDpKSkQGAUXAXKVUT8cyM7AbuBRIBTYAo5VSO0TkGuAuYJ5S6nMRCQaeBkqA85VSF5zoPNX0KCnjKWXF/k+saXVjtZbpx/EazCVGSSmlVgM5xy0eAOxVSiUppSqABcAIx/bfK6WGA7c63mcqpe4BHgOyT3QOEblDROJEJC4rK8tZP4pWi6qqAuLj/0F+/l8Aulho9VZdLHJylrNr1xSa01D/5saITu9w4GCN96nAuSIyGLgO8ACWAIhIFPAE4AP870QHU0rNAmaBvYXhpMzaCSil2LlzAnl5a/QcUdoZKyzcxKFD7+Pt3ZX27R8yOo52Ai4zSkoptQpYddyyZOAOA+JodZCS8l+ys7+hU6fXaNNmsNFxtCYuMvJfFBauZ9++R2nV6mz9O+WCjBgllQa0r/E+wrFMa0Jyclawf/+TtG17MxERU42OozUDIkJMzMd4eXVmx46bKStLNTqSdhwjCsYGoIuIRIuIOzAK+N6AHNoZyMj4BG/v7nTr9qGe3kFrMBaLHz17fo3VWszhw7ONjqMdx6mXpERkPjAYCBKRVOBppdRsEbkXWAaYgTlKqQRn5tAaXkzMJ1RW6ieqaQ3PxyeW/v3j8fLqZHQU7ThOLRhKqdG1LF+Co2Nba1rS02cREDAcT8/2uLu3NTqO1kx5e3cGoKRkL+XlKbRpc4nBiTRwoTu9NdeXnf0Du3ffSWrqdKOjaC3E7t13kJBwPaWlyUZH0dAFQ6ujsrKD7Nw5nlat+hId/bzRcbQWolu3D1DKxo4dN2GzVRgdp8XTBUM7JaWsJCbeilKVxMYu1Hflao3Gy6sT3brNprBwA/v36ylnjKYLhnZKqakzyM9fQ5cu7+Dt3cXoOFoLExx8A6Ghkzl48BXy8n4zOk6L5jI37mmuKzR0EmazN+3ajTU6itZCde78Ju7uYfj6nmN0lBZNFwytVlVVhYi4YbH4ERZ2p9FxtBbMbPYmOvoZAKzWEkwmL33/jwH0JSmtVrt3T2Hz5guw2SqNjqJpAJSXHyYurg/p6e8bHaVF0gVDO6GMjAVkZn5OUNAITCY3o+NoGgDu7sF4ekazb9/DlJTsNjpOi6MLhvY3ZWUH2bPnLvz8ziMy8gmj42jaUSImYmI+wmTyJDFxjG79NjJdMLRjKGVj584J2GyVxMTMw2TS3Vyaa/HwCKNr1/cpLNzAgQPPGR2nRdEFQztGZWU2VVU5dO78xtHpGTTN1QQH30BIyDjy8lZis1UZHafF0F8ftWO4uwdz9tnrEdG/Gppr69r1HUwmT/2Ux0akWxgaYL+b+8CBF6iszMNkctNDFjWXZzb7IGKmoiKLjIzPjY7TIuiCoQFw8OAb7N8/jZycpUZH0bR6SUl5mcTEMeTn/2F0lGZPFwyN4uKd7N8/jaCgkQQH32x0HE2rl6ioZ/DwiGTnzglYraVGx2nWdMFo4ZSysmvXJMxmH7p0eVdfitKaHIvFl5iYOZSW7iE5+Wmj4zRrumC0cGlp71JQ8AedO7+Jh0c7o+No2mlp0+YSxwSFr1FQsMHoOM2WSw2FEZHuwANAEPCLUupdERkJXAn4AbOVUsuNzNjcBAWNpKoql5CQMUZH0bQz0qnT/wDBw6O90VGaLVFKOfcEInOAq4BMpVTPGsuHAdOxP9f7Q6XUyzXWmYC5SqkxNZa1AV5VSk2q7Vz9+/dXcXFxTvgpmp/qf3d9CUrTNBHZqJTqf6rtGuOS1MfAsJoLxD5w+m1gOBALjBaRWMe6a4DF/P2Z39Mc+2gN4PDhT9iyZSiVlXlGR9G0BlVamkx8/BCKi3cYHaXZcXrBUEqtBnKOWzwA2KuUSlJKVQALgBGO7b9XSg0HbgUQu/8CPymlNjk7b0tQUZHFvn0PY7OVY7H4GR1H0xqU2exDUVE8u3bdgVI2o+M0K0Z1eocDB2u8TwXCRWSwiMwQkff5/xbGfcBQ4AYRmXL8gUTkDhGJE5G4rKwspwdvDvbtewirtZBu3d7HfvVP05oPd/e2dOr0GgUFazl06AOj4zQrLtXprZRaBaw6btkMYMZJ9pkFzAJ7H4YT4zULOTk/k5HxKR06PIWPTw+j42iaU7RrN56MjLns2/cvAgOvwcMj1OhIzYJRXy/TgJpDGSIcyzQnO3DgBby8uuppy7VmTUTo2vU9bLYyUlJeNDpOs2FUC2MD0EVEorEXilHALQZlaVF69fqB8vJUzGZPo6NomlN5e3eld+/F+PmdZ3SUZsPpLQwRmQ/8CXQTkVQRmaSUqgLuBZYBicAipVSCs7O0ZBUVGdhsFVgsvvj4dDc6jqY1ijZthmA2+2C1lmG1lhkdp8lzegtDKTW6luVL+PvQWc0JlFLs2HELNlspffuu1fdeaC1KVVU+Gzf2Jzj4VqKjnzE6TpOmh8i0AJmZC8jL+5WQkHG6WGgtjsXSGl/fc0hJeYmSkj1Gx2nSdMFo5qqq8tm37yF8fc8hLGyy0XE0zRCdOr2GyeTJnj334OzZLZozXTCaueTkZ6ioyKBr13f1k8m0FsvDI5To6OfJzf2Z7OxvjI7TZOmC0YzZbJXk5/9BaOgd+Pr2MzqOphkqLOwufHx6kZm5wOgoTZZL3binNSyTyY2zz/4Dm02PDnGWnNIcDhUcIrMgk+zCbHJKcvAx+XBe8HlUVVWxaO8i8srzsCorgmAWMx38OnB55OVYLBZWpK3AzeJGgE8Aga0CCfELIdQvFF8PX6N/tGbHZLLQu/cy3N1DjI7SZOmC0UwVFMTh5dURN7cAzGYfo+M0STabjZW7V7Lp4CaSjiSRnJ9MRkkGPsqHsa3GUlxczP8K/schDh2zXwfVgQlMAGAmM8mRY6dS66q6krPWvuxVXqVYio9Z39vUm0mtJ+Hj48PrOa/Tyr0VoT6hRPlH0TmoMwOjB3J25Nl6AMNpqL7ju6IiC6Uq8PAINzhR06ILRjNktRaTkHAt3t4xnHXWz0bHcWlKKfYe2cuaPWvYdHATCVkJ5JbkcrvX7eTm5jLPOo89sgcU+Iov/iZ/fDx88PHxITg4mMmlk1EWRYB3AAE+Afh7+RPkE0T71u2xWCyMt41HRLCYLCgUVbYqsIGHyYOqqioG5QwirzSP3JJcckpyyC3JxRdfwi3hFBYVQiUcLD3IttxtlKbZHz86cNVArnK/ilZtWvFW4Vt0ad2FHiE9OCfyHC7qchHtfPWDsE7GZqtg48azadWqL716fW90nCZFF4xmKCXlZcrLU4mN1ddqa1JKsT9nPysSV9DbrTeH0g8xfc90fiv97eg2/vjT3r09gUGBxMTE0NO9J2382xATGkPbgLaYzccOHBjJyDPKFBkZedL11S2ViooK0rLTSExLpLKkEvcSd/Zl7oMc+O3wbyw5vAS22Pe5xf8Wbul8C37BfqSb0hnafSiB3oFnlLM5MZncCQ+/j6Skf5GTs4yAgMuNjtRkOP0BSo1JP0AJysoOsH59DEFB1xEb+5nRcQy3N3Mv8zbMY3XyarbkbiHXmgvAvepeOrTqQJ5/HsVexQxoP4DzOp9HZEjk34qCq6uoqGB36m7W7l1LXGockZWRmI+Y2VaxjQVi/9IQ4R7B2UFn849O/2DsuWMJ9GnZBcRmK2f9+h6YTO70778Fk8nN6EiGqusDlHTBaGYSEm7myJEfGDBgF56eLe9RlWn5aSzYsICgsiAqD1eyIm0FC1mIL77EeMVwTsg5DOo0iCGxQwhsE9hs+wFsNhsHDh1gecJyfk/+nU3Zm9hXsY9yKed+8/3079Cf8sBy3ALcuK7vdS2ykz07+zu2bx9J584ziIi4z+g4hmqwgiEiIcCLQJhSarjjyXgDlVKzGyZqw2npBcNmq2THjpvw8TmrxUyBoJRizb41zFs/jxUpK0guTwbgci7nxogbCe0Qik+wDwNjBuLu7m5sWIMVFhWyYtsKPHI92L9/Px9mfUi8xGPGTGyrWK7sfCXjzxtPTEiM0VEbhVKKrVsvw9OzE926vWd0HEM1ZMH4CfgIeFIpdZaIWIDNSqleDRO14bT0glFNKVuzfjCS1WZlV9ouclNyid8RzwPpD2BTNqIsUVwUchEjeo5geN/heHl6GR3VpWXnZvPVhq9YsnsJf+X8RabKpA1tmNl5JrGxsfhH+BMVFNVsW2EAVmuZnrmZhi0YG5RS54jIZqVUX8eyeKVUnwbK2mBacsHIyVmBp2d7vL27GR3FKZRS/LLrF95d+y4/p/+Mn9WP27md0NBQioKLGNZnGLFRsc36w82ZrFYraxLWsH7netzT3TmSd4TX5XX83fy5puM13Df4PmLbxRod02mKixMxmdzx8upkdBRD1LVg1GWUVLGIBALKceDzgPwzzKc1IKu1mJ07x+Hl1Ym+fdcYHadBlZeX8/LSl3l7+9tkVWVhURZ6e/VmZJeR3DP4HgICAoyO2CyYzWYG9x7M4N6D7aPJDu4n/7d8lhxYwns73+O9Xe/R2bMz0wZOY+yFYzGZmk8L1motZfPmC/HzG0jv3j8aHcel1aVgPAR8D3QSkbVAW+AGp6bS6uXgwVepqDhEjx5ApOENAAAgAElEQVRfGh2lQZRWljJrzSza5LTh4K6DbKrcRIBbAJNiJnHnxXcS1S7K6IjNmojQMbIjM8bOYLqazvqd65n1xyx+SvuJVb+uIntDNoFdAmkd2ZoRZ43A1MQvf5rNXkRGPkpS0mPk5v5CmzZDjI7ksuo0SsrRb9ENEGCXUqrS2cFOR0u8JFVens66dV0IDLyCHj2+MDrOGdmevp2Xfn6J7w58R7Eq5lrztUw8ayJ9+vQhIiJCX24yWFVVFXv27CE+Pp6Zu2fyJ38SYglhTPcxPHrpowT7Bhsd8bRZrWWsXx+DxeJP//4bW9xEnQ3ZhzHuRMuVUnNPM5vTtMSCsXPnJDIy5jFgQGKTvf6aejiVaxdcS1x+HCZloo9XHyb3ncyEiyfg6aE7JF1RVm4W03+ezue7P2e/dT8WLIyMGMnHoz7Gx6dpTkWTkTGfxMRb6NbtI0JDJxgdp1E1ZMGYWeOtJzAE2KSUcrnLUi2tYCilSEp6FBELHTu+ZHSceimtLOXbuG+x7rOyd+9evjZ9TY+QHvxr6L/o08nlxlNotbDZbCyOW8wba9+gNL+UK8xX0KtXL8zRZq7qfVWTahUqpYiPv5jAwKuIjHzU6DiNymk37omIP7BAKTXsdMPVctyRwJWAHzBbKbVcRAYDzwEJjnOuOtkxWlrBaIpyS3J5ZskzfJz4MUXWIp70fpKh5w2lX79+TfabqWaXnZ3NunXr+G7zd3xg/YAI9wgePvdh7h18LxZT05iFqLkPSa9NXQvG6fzNFAPRdQwxR0QyRWT7ccuHicguEdkrIo8BKKW+VUpNBqYANzs2VUAR9pZN6mlkbbYKCtaRm/uL0THqLLMwk9s/v53wV8OZkTCDMHMYb5//Nk9NfYpBgwbpYtEMBAUFceWVV/LKA6/wWMxjVFZWMnXNVMJeDOP5n56nvKrc6IinVF0scnJ+pqIi2+A0rqcul6R+wDGkFnuBiQUWKaUeO+XBRQZh/8Cfq5Tq6VhmBnYDl2IvAhuA0UqpHY71rwGfKaU2iYhJKWVz3G3+ulLq1pOdr6W0MJSysXHjACorszj33D2YTK57B7PVaiU+Pp5vVn3Dy4Uv09urN09c+ATXn399k7pcodVfeUU5M5bOYOaWmRyxHuHl0JcZPmQ4nTp1cul/+9LSZNat60R4+H106fKm0XEaRUP2YVxc420VcEApVedv+yISBfxYo2AMBJ5RSl3ueP+4Y9OXHa+flVIrjjuGO/D5ifpNROQO4A6AyMjIfgcOHKhrtCYrM3MhO3aMIibmE9q1O+GYBMPll+bz2HePsS5pHddUXENERAQ9zu3BeT3Pc+kPC63hVVVVseyvZezesJvc/Fy+9vya8f3G89CQhzCbXHM00q5dkzl8+BMGDNiJl1dHo+M4nctMPniCgnEDMEwpdbvj/VjgXOytjvHYWxzxSqn3ROQ64HLAH3hX92HY5/Jfv747ZnMr+vff5HLD/8oqy3jup+eYET+DIlVEX8++zBsxj9hu+i7sls5qtbLsz2XcufJOUm2phLuF8/w/nmf8eeNd7nfDPly9M0FBI4mN/dzoOE53xn0YIlIoIgUneBWKSEHDxgWl1AylVD+l1BSl1HuOZV8rpe5USt18qmLRUqSnz6KsLImOHf/rcsVi2ZZltP9ve17c/CJh5jAWDl3Ixkc30iOmh8t9IGiNz2w2c8WFV5D0WBLP9X6O8qpyJi6fSI9Xe7AnfY/R8Y7h4RFGRMRUMjPnU1i4yeg4LqPWgqGU8lVK+Z3g5auU8juDc6YBNefdjnAs0+rAbPambdsbXeqhL4ePHOarr75i5Tcr8Vf+zBwwkx2P7eCmC27ShUL7Gzc3N6ZdO42kh5N4sNODVJRUMH/2fH799VeKSoqMjndUZOSjeHt3p7xcfzxVq/MlKREJxj5aCQClVEod94vi2EtSFuyXn4ZgLxQbgFuUUgn1CX4iLeGSlCtJzU1l8qLJxB+OZ4ppChddcBEXXHABHh4eRkfTmpC8vDx++eUX1m9bzwemDxjTfQyvjHwFTzfjb9pUSrWILz0NNqxWRK4RkT3AfuA3IBn4qY4h5gN/At1EJFVEJimlqoB7gWVAIvYRV2dcLJq7ysocDh36CJutyugoVNmqePL7J+kyowvLDy1nQOAA7rz7Ti655BJdLLR68/f35/rrr+fmW24myiOKmTtm0uGVDny23vgnRooINlsVWVlf05weNne66nIfxnPAecBupVQ09pbBX3U5uFJqtFIqVCnlppSKqH7oklJqiVKqq1Kqk1LqhdNO34KkpPyXXbsmUVq629AciWmJdHmlCy9ufpFot2iWj1zOd/d+R7vAdobm0pq+s7uezcZHN/L2eW+jrIoxP41h0MxBFJQ0eJdpvWRmzich4XqOHFlsaA5XUJeCUamUOgKYHPdFrARO2XTRGk55eTppaTMICbkVHx9jnklgtVr5448/+PLjL7FUWHiu93NsfXQrQ/romT21hiMi3H353ex5aA9jQseQcySHD9/7kF27dhmWKTh4FF5endm//0mUshmWwxXUpWDkiUgrYA3wmYhMx363t9ZIDhx4AaWqiIr6jyHnXxi3kG7/7cb3y7+nU3QnNty/gWnXTsNiaRrTPWhNT+tWrZl3xzyW3LYET09P3p7/Nme9ehabD25u9CwmkxtRUc9SXLyVzMyFjX5+V1KXgrESaA08ACwF9gFXOzOU9v/Kyg5w6NAHhIbe3ug3EB0pPsKw94YxavEoCqsKueDyCxg9ejT+/v6NmkNruSIjI7nzzjtpf1Z79hTvYcCcAUz9cipV1sbtywsOvhkfn54kJz/jEv2IRqlLwbAAy4FVgC+w0HGJSmsEFRWZ+Pj0IjLyyUY976K4RXR5vQvLDy/nurbXkfhgIlcPvLpFjBjRXIvZbOaf1/6T9ePX08uzF28mvEmP13qwNW1ro2UQMREV9SwmkwcVFemNdl5XU59htb2xTwp4PZCqlBrqzGCno7kOq23MoX02m421a9dy26+3USAFvDXkLW684MZGObemnYrVauXZb57llYRX6Gfpx+wbZtOtW+M8x97+Wama5Wy2DT41iIi0A24ERgG+SqneZxax4TW3gpGV9Q1t2gzBYjmT+yTrbvWe1Wz4bQOFqYVEdI1g5BUjCfIPapRza1p9xO+LZ9WyVeRl5hHcI5gRl44g3D+8Uc5dVZVPScke/Pyaz9ifhrwP424RWQX8AgQCk12xWDQ3xcU7SEi4npSU/zr9XDZl49FvHuWSzy9hbvpcRo4cyaTRk3Sx0FxWn059uPeOexk4cCDPJjxL9xndG+2+jR07RpGQcC1Wa1mjnM+V1KVt1R54UCnVQyn1TPU05JpzJSc/i9nsQ0TEVKee51D+IQa8OYD/bf0fvTx78dWkr+jTp4/uq9BcnsVi4fLLL+ejqz+ilbRizE9jGDt3LBVVFU49b0TEw5SXp3Lo0IdOPY8rOmXBUEo9rpSKb4wwml1xcQJZWYsID78Pd3fnfctfvWs1sTNiic+P576O97Hh4Q10Du/stPNpmjMM7zec7Q9sZ4jfED7d/ym9Xu9Fao7znrfWps0QWre+iJSUl1tcK6P59d40A8nJzzlaFw857RwJCQms+HIFbWnLV8O/YsbYGfq+Cq3JCvAL4OcHf+Y/Pf+DKlUsmruIw4cPO+VcIkJU1DNUVKRx+PBsp5zDVelPCBdjs1VhtRY4rXWRV5LHXZ/dRae0TkS3jybuxjj8/BqnU13TnElE+Pf1/2bCORP48ssvmf7BdCpjK3nlulcwNfDIJn//f9C69UUtbupzpz9AqTE1p1FSSlkb/HkXm1I2cdW8qzhceZhnuz3L4zc9jtnsWs/U0LSGUFxczKTZk1iYu5ABrQfw0+SfCPAJaNBzWK0lmM3eDXpMozTYKCmt8ZSVHaCszP6I2YYuFp+t+4wLPrqA/Kp8Zl00i2mjp+lioTVbPj4+fHbPZ9wTfQ9xeXH0fLMnW1K3NOg5qotFaWlyi+nL0AXDhSQlPUZc3NkN/sv33I/PMXbpWAJMAawavYrbh9zeoMfXNFdkNpt5a9xbfDjoQ/Kr8hk4eyA/xP/QoOcoKdnF+vVdWkxfhi4YLqKkZBeZmQsJDZ2M2dwwD45RSrFq1Soy4jI43+d8Nt27iXO6ntMgx9a0pmLiJRP55eZfiDZH8+ePf7Jz584GO7aXV1d8fQeQkvJfbDbnDud1BbpguIgDB17EZPKkffuGGRmVWZjJxFkTWblqJUP7DOW3qb8R0iakQY6taU3NeTHnse7BdUSHRPP5ws/556J/NsgEhiJChw5PUV5+kMOH5zZAUtemC4YLKC1NIiPjM8LC7sTdPfiMj5eQnkDvGb359NCndDy3IyNGjND9FVqL16pVKyZMmEBheCGvJr7K+TPPp7Cs8IyPGxBwOb6+/UlJeanZz2TrMgVDRDqKyGwR+bLGskgR+VZE5ojIY0bmc6b8/N8xm71o3/6fZ3yslbtWMvDDgRRUFTBn8BzGDx+v79rWNAc3Nzden/Q693W8j7i8OPq82Yf0vDObfdbeyphGeXkKhYXrGyipa3JqwXB80GeKyPbjlg8TkV0isre6ECilkpRSk447RC/gS6XUbUBfZ2Y1Urt24xg4MBUPj7AzOs78DfMZtmAYZmXmx2t/ZNzgcQ2UUNOaDxFhxtgZvNL/FVLKU+j7Vl+2p20/9Y4nERh4Deeem0Tr1uc3UErX5OwWxsfAsJoLxD5e9G1gOBALjBaR2p47+hcwSUR+xf7wpmanvDwNAIul9RkdZ//+/SxdtpQQUwirx6/mkt6XNEQ8TWu2HrnqEeZeMpcKawVzF80lLy/vtI8lInh6tgegqurML3O5KqcWDKXUaiDnuMUDgL2OFkUFsAAYUcshJgJPK6UuAa480QYicoeIxIlIXFZWVkNFbxQVFRmsW9eZgwdfO6PjLF6/mE8//ZRzA85l6/1b6RXVq4ESalrzNvqi0cSNjyOgPIA5c+awOenMHgG7e/fdxMcPojndEF2TEVODhAMHa7xPBc4VkUDgBaCviDyulHoJe6viGRG5BUg+0cGUUrOAWWC/09uZwRtaaup0bLZyAgNP74m3SinuXXQv7+58l0eCHuGRiY/g5eXVwCnPTGVlJampqZSVtYwbm7RT8/T0JCIiAjc3N6OjANApqhMTJ07kvjn38di8x5h9+WzGnjf2tI7l6zuA9PR3yclZSmDg8AZOajyXmUvK8djXKcct2w7cYEwi56qszCMt7W3atr0Bb++u9d5fKcX4T8czL2ke5/ucz9O3Pe1yxQIgNTUVX19foqKidOe7hlKKI0eOkJqaSnR0tNFxjgoJCeG5sc/xx0d/MGHZBIoripkyaMqpd/zbcW4lOflpUlJebJYFw4hRUmnYn7FRLcKxrEVJT38Hq7WAyMjH672vUorRH49mXtI8hrYeysoHVuLj5eOElGeurKyMwMBAXSw0wH6tPzAw0CVbnN0iuvHnlD/pYOnAPSvv4a2Vb9X7GCaTG+3bP0J+/u/k5a1xQkpjGVEwNgBdRCRaRNyxP/L1ewNyGEYpG4cPf0RAwHB8fes/+OvV715lYcpChrcZzk/3/oS7m7sTUjYcXSy0mlz596F92/asvWstHS0deWD1A6yMX1nvY4SGTsLNrS2pqW86IaGxnD2sdj7wJ9BNRFJFZJJSqgq4F1gGJAKLlFIJzszhakRM9OsXR5cu79R73zVr1lAUX8S0DtP44Z4f9DMsNK2BhQaEsvbutdzR+g7W/riWffv21Wt/s9mbnj2/JSZmjpMSGsfZo6RGK6VClVJuSqkIpdRsx/IlSqmuSqlOSqkXnJnB1ShlRSmFxdIaL6+oeuynGP/JeD775TPO6n0W/xn3H333dgN49dVXiYmJoU+fPpxzzjnMnWuf3iEqKors7GyD02lGCW4TzGt3vEZQUBDPf/4803+dXq/9W7c+/4yHyrsil7nTu6XIyJhPXNxZlJcfqvM+SilGfzKauclzORJ8hJEjR2Iy6X+6M/Xee+/x888/s379euLj4/nll1+a7XBIrf68vb0ZO3Ys6y3rmbpmKu/8Vr8rAoWFm4iL60dpaf1aKK5Mf+o0IqUUBw++glI23N3rPhHgbZ/dxsID9j6LBXcu0MWinpKTk+nevTuTJ0+mR48eXHbZZZSWlvLiiy/y7rvvHn3ioJ+fH+PHjz9m39LSUoYPH84HH3xgRHTNYD4+Pvw65Vc6Wjpy36r7mL227tOYu7uHUly8/Yzvs3Il+gJ4I8rJWUpx8TZiYj5G6vjIyKlfTuXjfR9zSetL+P7u75v0ZailS5c2+HOW27Vrx7Bhw0653Z49e5g/fz4ffPABN910E1988QWFhYV07Nix1n2KiooYNWoU48aNY9w4Pc1KSxXSJoQ1d61h4DsDmbJiCv5e/lx/9vWn3M/DI5R27cZx+PBHREU90yATixpNf1VtRCkp/8XDI4Lg4NF12n5H4g5+TPiRAd4DWHL3Et3BfQaio6Pp06cPAP369SM5OfmU+4wYMYKJEyfqYqERGhDKr5N+JcgUxGs/vcaRI0fqtF/79o9gs5WTljbTyQkbh/4EaiQFBevJz/+NTp1ew2Q69TDY/fv38/VXXzM1dCpjx4zFw92jEVI6V11aAs7i4fH/f39msxmbzUarVq1ISkqqtZVxwQUXsHTpUm655RaXHgqqNY6OoR1ZPXE1X332FXPnzmXSpElHL2fWxtu7G0FBI0lLe5v27f+FxdKqkdI6h25hNJJWrfoQE/MJoaGTT7ntVxu/4h/z/oGltYWJYyfi6+3bCAlbnscff5x77rmHgoICwH4JqnqUFMCzzz5LmzZtuOeee4yKqLmYLhFdGD9uPOml6Qx4awD7sk7dod2hw1N07vwGJlPT/9KnC0YjMZncadduHBbLyT/8f9/zO2N+HEO5lDP6ptEuOd1Hc3HXXXfxj3/8g3POOYeePXty0UUX/W1AwfTp0yktLeXRRx81KKXmakJDQxl02SCSK5MZ/MFgsotOPvza17cv7dqNx2RyjbmzzoQ0p2GE/fv3V3FxcUbH+Jvk5OewWFoTEXH/SbfbnbGbAe8PsD+Le+wq+nZs+o8ASUxMpHv37kbH0FxMc/i9+OCXD5iyZgrdfboT92Acnm6etW5rs1WQljYTL69uBAVd1Ygp60ZENiql+p9qO93CcLLKyiOkpLxEUdGWk253pOgIl8y+hHJVzqIRi5pFsdC05mzykMk81espEkoSuGLWFSe9h0fEQnr6LJKTn27S9/roguFkaWnvYrOV0r79Q7Vuo5Tiy8VfUlFRwYwLZ3B5n8sbMaGmaafr6eueZlz4OHZl7WL1utW1bidion37hykq2kRe3m+NmLBh6YLhRFZrGWlpMwkIuAIfnx4n3EYpxerVqzmUeIj5F89n8pBTd4prmuYaRIQ5E+fwQucX+G35byQlJdW6bUjIWNzc2nLw4KuNmLBh6YLhRBkZn1JZmUn79o/Uus0T3z3B/SvvJ7ZXLJcM1o9V1bSmxmw2M/rG0bQKbMXVn13N4q2La9nOi/Dwe8nJWUxx8Y5GTtkwdMFwIm/vroSF3Y2//+ATrl8Ut4hXtryC2dvMyGtG6rH+mtZEeXh4cN0N15GhMhj97Wj2ZO454XZhYXcTEDAMm62ikRM2DF0wnMjffxBdu759wkKwPW07E5dMJNgUzOLbF7v8My00TTu5qJAoFoxYQLkq5/I5l1NcXvy3bdzdg+jd+yd8ffsYkPDM6YLhJOnpH1JWlnrCdfml+Qz7ZBhKKb668StCA0IbOZ1WU2FhYYMfs6ioqMGPqbm+oWcN5cX+L7K/fD8jZ4+sdURUeXkaubm/NnK6M6cLhhMUFW1n9+7JZGTMO+H62d/PJqcihzcveJPzY85v5HRaTSUlJcTHxwOwceNGfv/9dwD27dvHihUr6nSM6v0+/fTTo48edXNzY9WqVU7JrLm2h698mFvDbuWvzL/4ZcMvJ9xm9+4p7NhxCzZbeSOnOzO6YDhBauqbmExehIXd+bd1W7dupSCxgM8HfM4dQ+8wIJ1W09dff83AgQMB2L59OwUFBSxcuJDFixeTlJTEBx98wGuvnXx66ur90tLSWL58OWC/pp2bm+uU1ovm+j6a+BEvhL/Aup/XkZWV9bf14eEPUFmZQUbGfAPSnT5dMBpYRUUmGRmf0q7dBNzcAo5Z98OWH3jsu8eIjIzk6suvNiihVlNGRsbRWYAPHTpEXFwcN998M2VlZeTn5zN58uRjpgv59ttvmTBhAhMmTGDhwoXH7PfAAw8cMwvuxRdfzDfffNOoP4/mGtwsbkwcNRGzm5nbPrqNgzkHj1nfps0QfHx6kZr6epO6kc9lZqsVkY7Ak0BrpdQNtS1zdenp76JUORERDxyzPCUnhTHfjcENN2aPmN2kn2vRXGRkZODt7X30vbe3Nx06dGDr1q24u7tjtVoBjikYI0eOZOTIkcccp3q/3bt3ExUVdXR5QEBAvZ8HrTUfvr6+9L+sP0999xRXzrmSTVM3YTHbP3JFhIiIB9m1axJ5eb/Sps0Qg9PWjVNbGCIyR0QyRWT7ccuHicguEdkrIo8BKKWSlFKTam53omWurrIym6CgkXh7dzu6zGqzctWcqyhRJcy9ai6hgbqT2xWkpaURGBh49P3999/P6NGj6d27N1dffTW9evUC4IEHHqjtEMfst3XrVi677LJj1pWXN61r1FrDGtpnKI90f4RtxduYsmDKMeuCg2/B3T2MwsJNBqWrP2dfkvoYOOYhCCJiBt4GhgOxwGgRiXVyjkbTpctMevT46phlD3zxANuKtzG121SuOPsKg5Jpx6uoqMDN7cQziHbq1OlvH/6nMmbMGDw9j52ArildbtCc48UbX+RC3wuZs3cOX2/6+uhys9mTc8/dQ2TkPw1MVz9OLRhKqdVAznGLBwB7Ha2HCmABMOJ0zyEid4hInIjEnahzqbEopSgp2ePI9P9/rduSt/HezvcY4DOAl258yah4Ld51113HtGnTGDRoEJGRkaxYsYLg4GByc3OPbiMiDfoCcHfX99e0dCLClxO/JFACuXPxnZSUlRxdZzbbL4lWVtbtCX5GM6LTOxyo2QOUCoSLSKCIvAf0FZHHAU607HhKqVlKqf5Kqf5t27Z1evja5OevZf36rmRnf3d0WVVVFb//9DuTPSbz1YSvdL+FgbZt24a/vz+rV69m+vTpfPbZZ0RHRx8zgkUpVa/Xb7/9xs0331zr+vLycoz8ndRcR0ibED6+8mOus13HiuXHDtdOSXmFv/7qSFVVgUHp6s5lOr2VUkeAKada5qrS0mZgsfjTps3Qo8s+/O5DMjIyeOjWh4gIijAwnevYvHnw35YFB99EePjdWK0lbN3690t27dpNIDR0AhUV2SQkHDv2oW/fVac8Z0lJCfn5+UydOhWAyspK/P39EZFaH7E5a9YsRISCggIefvjhE24zaNAg/vjjj1rP+9dffzF8+PBT5tNahiv7XYlnnidr1qzBO9SboefYPyv8/QeTlPQvDh/+hIiI+wxOeXJGtDDSgPY13kc4ljVZZWUHycr6mtDQyZjNPgC8ueJN7tp2Fx7dPejSpYvBCVu2HTt20K9fv6MtvK1bt9KzZ08ARowYwbp16/62z/FDak80nPZUcnNz9b+9dozBgwez2383Vyy5gk0H7J3dfn4D8PM7j7S0mShlMzjhyRnRwtgAdBGRaOyFYhRwiwE5Gkxa2tuAIjzc/uznvZl7eWLtE0S7RTP12qnGhnMxJ2sRmM3eJ13v7h5UpxbF8bZt20afPv8/d8/WrVsZMcLebRYWFkZlZeXf9qnuDK8uGCcaTrtt2zbWrl3L8uXL/9ZBnp2dzaWXXlrvrFrzZjabeeLaJ/j242+5cf6N7PznTtzMboSHP0Bi4mhycn4iMPBKo2PWytnDaucDfwLdRCRVRCYppaqAe4FlQCKwSCmV4MwczqSUjczMzwkKGomnZweUUtww7waqqGLudXPxdK/9sY1a4zi+YGzfvv1oCwOgQ4cOf9vnwQcfBE4+pLZXr1788MMPJxxNFRQUhI+Pz5nE1pqpnh16Mq3vNJLKk7h/kf2xzW3bXo+7exjp6e8ZnO7knNrCUEqNrmX5EmCJM8/dWERM9O+/BavV3mH17x/+zZaiLUztMpULu19ocDoN4PXXXz/m/ckecqNpjeHxqx/nh90/MGv3LG5IvIEh3YfQo8eXx9y/5Yr01CBnoHo0jJtbGzw9O1BcXMy27dvo69mX/938P6PjaZrmokSEBeMWECRBfPPrNyilaN164N+mE3I1umCcgYKCP9m48RxKSnYBsHTpUvpZ+7F04lI9hFbTtJPqENyBJVcvISgriLi4OADy8n4nPn4IVVWuOWmlLhhnIC3tLUpL9+LhEcGbP7/J/G3zueiiiwgODjY6mqZpTcDZfc6mY8eOvLjsRdYlrUPEQl7er2RkfGp0tBPSBeM0lZcfIivrC0JDb+NgbhZP/PEE8e7xXHDhBUZH0zStiRARLrz0QpZZlzH2i7F4+/TD17c/aWlvueS0MrpgnKZDh2ahVBVhYXcz5vMxVFDBnBFzcLOceG4iTdO0E+kY2pFHez/KnrI9TPthGuHh91JSsoO8vJVGR/sbXTBOg81WSXr6+wQEDOPrrX+xNnctt0bcyqAeg4yOpmlaE/TUNU/Rw6MH07dOp9g0ADe3INLS3jI61t/ognFaFNHRzxMS9jAPrXiIttKWmaNmGh1K07Qmymw2M2vkLKqo4vaFdxMV9R+XvIHPZeaSakpMJndCQ2/jr7/+4uKqi7ni4ivw8znxnESapml1cX7M+dzZ4U7yD+QjMpLQ0DCjI/2NbmHUU0nJLlJTZ5CXd4iVK1dyTadrmDB4gtGxNE1rBl4b9Rp9fPqwePFiKivzSU9/H5utwuhYR+kWRj2lpb1FevospsV/hm9lIPOumOV5UUcAAA//SURBVHf02QeapmlnwtPTkyGXDuH+b+5nx9dLGBHyLRbL/7V379FR1FkCx7+XkPAICIZoZBMghI2MWdmJEEVHjdGZUVBccGVE0aOMLCNHEGcQBZx9DDN6GB5nEVwki4aH6IiuuqjxgcAwoscBAQPCBiGRUZLwCiiBEENouPtHN7HzIpX0o5LO/ZyTA1VdVb9b/Jq+qV/9+taFXHzxXW6HBtgVRpN4POUcPLicnd9fySfHPiOpT1KNR3waY0yg0v8xnfLO5Sz68n1O0ouSkufcDqmaJYwmOHz4Zao8J5iVt4c4iWPuyLluh2Qi2LvvvsuMGTN46aUfvsSlquzZs6dZx9uwYQN333139XJJSQknTrTMbxS3ZSLC/GHzOa2nyd4dS1nZR5w82TLqs1rCcEhVKSl5jjWlCRRVlfLEwCfsRncEacpjV8PltttuY8qUKRQXF1ev++CDD0hOTm7W8TIzM2tU7U1MTGTdunWBhmlCIDMtkyHxQ3j34G6+rmhPSckit0MCLGE45vGUcVYuYPGe46REpzBl6BS3QzJBcvz4cWbPnu3osazhdPbsWebMmcP48T88dLK4uNjRc8J37NjBvn37Gt2uXbt2HDp0KKA4TWgs/MVCOtCBRQUXUFn5dYv45rclDIeio7tzpmoWd3pGM//m+VZcMIK8//77LfJRqrNnz6asrKz6iYBFRUV07doV8A4n3XjjjcybN49Ro0bV2Xfr1q11yrj7P/DpnIyMDNasWRPCszDNlZyQzPS06Qw6NpK4uOwWMbnGZkk54PEc5+TJb9mwYQNZqTcyLGOY2yGZINq5c2e9H7pumzZtWo3lkpKS6kkW27dvZ/To0YwbN4577723epv8/HwWLFjA7t276dq1KytXrmTGjBkkJCRUP/DJX3x8PCUlrfoJyRFt2vBpLPhmAR9++CH33TecmBh3J9nYFYYDBw4sYdz/pPDm6Ze56aab3A7HBJHH46ke4tm7dy9jx45l5MiRLkdVv06dOnHq1CnAmzCuv/56gBq/eaalpZGdnc0DDzzA5MmTyc7OJiEhoXq72j+VlZV06tQp/CdjHImJiWHgTway8ODv+eNbCZw4sdXVeFp8whCRWBHZIiKu/FqvquR99Sz/u1/pcsFF9OzZ040wTIh89NFHZGZ6a4ClpKSQk5PjckQN69evX/XVQGFhIZdeeilHjhzhkksucbR/bm4uv/vd71ixYkX1PZl9+/Zx2WWXhTJsE6BrM66l5EwFS74+S1Gxu1Nsw54wRGSJiBwWkZ211g8Rkd0iUigi/tfiU4HXwhvlD8rKNpCzZy+qwpxh9hS9SPPJJ59w3XWt41G6Xbp0ITraWw05JyeHdu3aER8fz9y5dad3jxkzhqysrBrr6pt1tW3bNm644YaQxm0C0yGmA7++YjLfVCiv7HgJj6fMtVjcuMJYBgzxXyEiUcBCYCiQBtwjImki8nMgHzgc7iDP+XTXbN47CD+L+ymD+g1yKwwTIh6Pp1VNYMjKyqKgoKBZ+9aedVVRUUGfPn0czboy7npsyGMktY9n2ddVFO9f7locYU8YqroB+LbW6quAQlXdq6pVwEpgOJAFXA2MBsaJSJ14ReRXviGrLaWlpUGN1eM5wcIvPkAQ5g6fF9Rjm/Dav38/EydOrLFu27ZtNb6XcPToUcaPH09eXh4zZ85s8FgVFRWuTXFMSUlxPARVW+1ZV2fOnKm+D2Jatuj20Uwe+CTF38N/b2z4vRlqLWWWVCJQ5LdcDAxW1YkAIjIGOKKqZ2vvqKqLgcUAGRkZQf1fXFkppB8by+V9ori89+XBPHSblbUsq866u/7hLh6+8mEqTldw68u31nl9TPoYxqSP4UjFEUa+VvOG9F/G/MVRu9u3b+e9996joKCA1NRUAFavXs2ECROqt+nRowfZ2dmNHislJYWNGzc2+wt0gTo3tbapas+6au5xjDsm3TyJT3evIv67waiqK9NsW/xNbwBVXaaqueFud9OmTcRUJvLEsKfC3bQJsqFDh3L//fezatWq6nXl5eV06dKlycfKy8ujV69e1ctOvyXeln9M4KKiovht1nxOHOjMV1995UoMLeUKowTo5bec5FvnmsLit3jyi/v4RcojVmAwiM53RdA5uvN5X4/vHO/4iqI+I0aMYMKECTz++OMUFRXRu3fvGq+f74PNfwiq9ky5lvANXNM2DBgwgBc/msnkNdfyRp89REd3C2v7LeUKYzOQKiJ9RSQGuBt4282AZv35cT47Xs6AAde4GYYJovT0dA4ePMihQ4fIzc1l2LCaM7VbSkkQYxoSFRWFJLTnncOHWfnp1LC378a02leAvwL9RaRYRMaqqgeYCKwGdgGvqapr5RmPlRfxRnEBV3SN45b0290Kw4TA7bffzjvvvMP+/fvrXCmsWrWKcePGMWrUqBrlMxoT7KqyzdHUSrS1Y7YquK3Hb297lgvat+O57S+GvW03Zkndo6o9VTVaVZNUNce3/j1VvVRV+6nq0+GOy9+c1eP47jQ8nP6om2GYEBg+fDjLly+ne/fudV4bMWIEzz//PNnZ2bz66quOjxnsqrLN0dRKtLVjtiq4rUdc1ziG9xzExu++Z/3OpWFtu6UMSbUYZ86cYUXhWvp1juHBrH91OxwTZJmZmeTn55+32OBTTz1VY/ZUYwKpKtuQBx98sN71TqvQwvkr0daO2T/ehtpuSvtWBTe0/u2WxcS0g1kf/3tY27WEUcvefXu4skMfHuo/mnbt7J8n0kRFRTF79mzS0tLqvKaqTJ06laFDhzJw4EDHxwykqmx9Kioq6NatG+vXr2fq1KlUVlZWv1ZfFVpoeiVa/5j94z1f2w21b1Vwwy81MZ27el5G3OmuVFWF75nfLWWWVIuxa0cBgyrGMumWx9wOxYTI2LFj613/7LPPsnbtWsrKyigsLKxxxXA+zakqu2PHDqZPn15jvyVLlnDxxRfz+eefk5eXR//+/Zk1axZw/iq0QKOVaDdt2sS2bdt46KGH6sS8cePG6njra7ux9q0Krjue+vn7LF26lPz8/BpDgqFkCcPPge/28s7eP3LLZY/QoUMHt8MxYTZp0iQmTZoU8HE6derEkSNHAG/CuOOOO4Ca03YHDBhAbm79Xy3avHkzgwcPJjY2tnrduSq0y5YtIzk5uU6dqPqmBJeVlVVXoh08eDCDBw9uNN762m6s/cbaNqHRu3dvuvXozJ82/4H09DfC0qaNufiZt3YiL5T9FelZ3PjGxjQg0Kqy+fn5zJw5ky+//JKPP/7Y0T6BVKL1jzfcbZvmExH+FpfLnP1vsm7HS43vEAxOHkvZWn4GDRqkgbh0TgdNntVePR5PQMcxP8jPz3c7BFfk5OSEvc3y8nKdOXNm9fKKFSv01KlTjvYNNN6mtt1W3xfBtqd4k0bNQO9cnBrQcYAt6uAz1q4wfD7ctpQ9J09xe9I1rap6qWmZAqkq2xyBVqK1KritU2riVVzd/ULWHi7gZGXoy55bwvDJ/uxp2gs8euMzbodiIkAgVWWbI9BKtFYFt/W650f3UuaBRet+E/K2RCOo7EFGRoZu2bKlyft5PB5+/EwXenSIZcMjR0MQWdu1a9cuG8s2ddj7IniqTn9P4txYbriwH6+Pb95VoohsVdWMxrazWVLAqVOn+H3fl+iXluh2KMYY0yQx0Z14+SdvIlWxIS97bgkDiI2N5c47Rza+oTHGtEA33zAiLO3YPQxjjDGOWMIwIRdJ98lM4Oz90HpZwjAh1bFjR44ePWofEgbwJoujR4/SsWNHt0MxzWD3MExIJSUlUVxcTGlpqduhmBaiY8eOJCUluR2GaQZLGCakoqOj6du3r9thGGOCwIakjDHGOGIJwxhjjCOWMIwxxjgSUaVBRKQU+CaAQ8QDR4IUTmvR1s65rZ0v2Dm3FYGccx9VvaixjSIqYQRKRLY4qacSSdraObe18wU757YiHOdsQ1LGGGMcsYRhjDHGEUsYNS12OwAXtLVzbmvnC3bObUXIz9nuYRhjjHHErjCMMcY4YgnDGGOMI5YwABEZIiK7RaRQRKa5HU8oiEgvEVkvIvki8n8i8qhvfZyIrBGRAt+fF7oda7CJSJSI5IlIrm+5r4hs8vX3qyIS43aMwSQi3UXkdRH5UkR2icg1kd7PIvIb3/t6p4i8IiIdI62fRWSJiBwWkZ1+6+rtV/Fa4Dv3L0RkYDBiaPMJQ0SigIXAUCANuEdE0tyNKiQ8wGOqmgZcDUzwnec0YJ2qpgLrfMuR5lFgl9/yLGCeqv498B0w1pWoQmc+8IGq/gj4Md5zj9h+FpFEYBKQoaqXA1HA3URePy8DhtRa11C/DgVSfT+/AhYFI4A2nzCAq4BCVd2rqlXASmC4yzEFnaoeUNXPfX8/gfdDJBHvuS73bbYcCM+zHsNERJKA24AXfMsC3AS87tskos5ZRLoBmUAOgKpWqeoxIryf8Vbe7iQi7YHOwAEirJ9VdQPwba3VDfXrcOBF9doIdBeRnoHGYAnD+6FZ5Ldc7FsXsUQkGbgC2AQkqOoB30sHgQSXwgqVZ4AngLO+5R7AMVX1+JYjrb/7AqXAUt8w3AsiEksE97OqlgBzgX14E0UZsJXI7udzGurXkHyuWcJoY0SkC/AG8GtVPe7/mnrnWEfMPGsRGQYcVtWtbscSRu2BgcAiVb0COEmt4acI7OcL8f5G3Rf4OyCWukM3ES8c/WoJA0qAXn7LSb51EUdEovEmi5dV9U3f6kPnLlV9fx52K74QuBb4JxH5Gu9Q4014x/e7+4YuIPL6uxgoVtVNvuXX8SaQSO7nnwF/U9VSVT0NvIm37yO5n89pqF9D8rlmCQM2A6m+GRUxeG+Wve1yTEHnG7vPAXap6n/6vfQ28IDv7w8Ab4U7tlBR1emqmqSqyXj79c+qei+wHhjp2yzSzvkgUCQi/X2rfgrkE8H9jHco6moR6ex7n58754jtZz8N9evbwP2+2VJXA2V+Q1fNZt/0BkTkVrxj3VHAElV92uWQgk5ErgM+Bnbww3j+k3jvY7wG9MZbGv4uVa19Y63VE5EsYIqqDhORFLxXHHFAHnCfqp5yM75gEpF0vDf5Y4C9wC/x/nIYsf0sIjOAUXhnA+YB/4J3zD5i+llEXgGy8JYxPwT8B7CKevrVlzj/C+/QXAXwS1XdEnAMljCMMcY4YUNSxhhjHLGEYYwxxhFLGMYYYxyxhGGMMcYRSxjGGGMcsYRhTIiJSLJ/hVFjWitLGMYYYxyxhGFMGIlIiq8o4JVux2JMU7VvfBNjTDD4ynWsBMao6na34zGmqSxhGBMeF+Gt8/PPqprvdjDGNIcNSRkTHmV4i+Rd53YgxjSXXWEYEx5VwB3AahEpV9U/uR2QMU1lCcOYMFHVk76HOq3xJY2IK6NvIptVqzXGGOOI3cMwxhjjiCUMY4wxjljCMMYY44glDGOMMY5YwjDGGOOIJQxjjDGOWMIwxhjjyP8DrP/PF7tkYzcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_nckest(n, ests, omitted=0):\n",
    "    k = np.arange(0,n+1)\n",
    "    \n",
    "    points = tuple(j for j in nCk_iter(n))\n",
    "    plt.plot(k, points, \"k-\", alpha=.5, label=\"nCk\")\n",
    "    \n",
    "    for i, func in enumerate(ests[omitted:], omitted):\n",
    "        points = func(n,k.astype(np.float64))\n",
    "        plt.plot(k, points, est_styles[i], label=est_labels[i])\n",
    "    \n",
    "    plt.title(\"nCk & Estimates (n = \" + str(n) + \")\")\n",
    "    plt.xlabel(\"k\")\n",
    "    plt.ylabel(\"value\")\n",
    "    plt.legend()\n",
    "    plt.yscale('log')\n",
    "    plt.show()\n",
    "\n",
    "plot_nckest(2, nCk_ests)\n",
    "plot_nckest(3, nCk_ests)\n",
    "plot_nckest(10, nCk_ests)\n",
    "plot_nckest(100, nCk_ests)\n",
    "plot_nckest(100, nCk_ests, omitted=2)\n",
    "plot_nckest(120, nCk_ests, omitted=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'nCk_ests' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-109-f95eb5637f64>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     15\u001b[0m     \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     16\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 17\u001b[0;31m \u001b[0mplot_nckest_relative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnCk_ests\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     18\u001b[0m \u001b[0mplot_nckest_relative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m10\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnCk_ests\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     19\u001b[0m \u001b[0mplot_nckest_relative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnCk_ests\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'nCk_ests' is not defined"
     ]
    }
   ],
   "source": [
    "def plot_nckest_relative(n, ests, omitted=0):\n",
    "    k = np.arange(0,n+1)\n",
    "    points_true = np.array([j for j in nCk_iter(n)])\n",
    "    plt.plot(k, np.ones_like(k), \"k-\", alpha=.5, label=\"unity\")\n",
    "    \n",
    "    for i, func in enumerate(ests[omitted:], omitted):\n",
    "        points = func(n,k.astype(np.float64)) / points_true\n",
    "        plt.plot(k, points, est_styles[i], label=est_labels[i])\n",
    "    \n",
    "    plt.title(\"Error Factor for nCk Estimates (n = \" + str(n) + \")\")\n",
    "    plt.xlabel(\"k\")\n",
    "    plt.ylabel(\"error factor\")\n",
    "    plt.legend()\n",
    "    plt.yscale('log')\n",
    "    plt.show()\n",
    "\n",
    "plot_nckest_relative(3, nCk_ests)\n",
    "plot_nckest_relative(10, nCk_ests)\n",
    "plot_nckest_relative(100, nCk_ests)\n",
    "plot_nckest_relative(100, nCk_ests, omitted=2)\n",
    "plot_nckest_relative(120, nCk_ests, omitted=3)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}