multipartitions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multipartitions #"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook computes values of $p(a, b)$, that is, the number of partitions of Gaussian integers $a + bi$ into parts $u + vi$ so that $u > 0$ and $v > 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References ##\n",
    "* M. S. Cheema and T. S. Motzkin, \"Multipartitions and multipermutations,\" Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).\n",
    "* The Online Encyclopedia of Integer Sequences (OEIS) entry for p(a, b): [A090806](http://oeis.org/A090806)\n",
    "* Main diagonal: OEIS [A108469](http://oeis.org/A108469)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "\n",
    "from multip import timed, memoize\n",
    "from fractions import gcd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Recurrence ##\n",
    "\n",
    "This is equation (3.1.3) in Cheema and Motzkin which defines the recurrence used in this notebook. For the purposes of this notebook, $s = 2, U(n_1, n_2) = p(n_1, n_2)$. The outer sum is over $k_1$ and $k_2$ and the inner sum is over $t$ such that $t$ divides $\\text{gcd}(k_1, k_2)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\displaystyle (3.1.3)\\hspace{2pc}n_1U(n_1,\\ldots,n_s)=\\sum_{k_i>0}\\big(U(n_1-k_1,\\ldots,n_s-k_s)\\sum_t k_1/t\\big);$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@memoize\n",
    "def gcd_(a, b):\n",
    "    return gcd(a, b)\n",
    "\n",
    "@memoize\n",
    "def inner(k1, k2):\n",
    "    D = gcd(k1, k2)\n",
    "    return sum([k1//t for t in range(1, gcd_(k1,k2)+1) if D//t == D/t])\n",
    "\n",
    "@memoize\n",
    "def p(a, b):\n",
    "    if b > a:\n",
    "        return p(b, a)\n",
    "    if(a==0 and b==0):\n",
    "        return 1\n",
    "    if(b==0):\n",
    "        return 0\n",
    "    if(a==1 or b==1):\n",
    "        return 1\n",
    "    if(b==2):\n",
    "        return a//2 + 1 # Exercise: prove this!\n",
    "    return sum([p(a-k1, b-k2)*inner(k1, k2) for k1 in range(1,a+1) for k2 in range(1,b+1)])//a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0:00:17.097694\n"
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       "      <td>...</td>\n",
       "      <td>92802791172</td>\n",
       "      <td>102581140543</td>\n",
       "      <td>113295193417</td>\n",
       "      <td>125025773840</td>\n",
       "      <td>137860045355</td>\n",
       "      <td>151891803561</td>\n",
       "      <td>167222083347</td>\n",
       "      <td>183959492143</td>\n",
       "      <td>202220888315</td>\n",
       "      <td>222131761469</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>1</td>\n",
       "      <td>11</td>\n",
       "      <td>53</td>\n",
       "      <td>184</td>\n",
       "      <td>507</td>\n",
       "      <td>1217</td>\n",
       "      <td>2622</td>\n",
       "      <td>5231</td>\n",
       "      <td>9780</td>\n",
       "      <td>17404</td>\n",
       "      <td>...</td>\n",
       "      <td>180523195041</td>\n",
       "      <td>200145723313</td>\n",
       "      <td>221711050361</td>\n",
       "      <td>245393563798</td>\n",
       "      <td>271381601376</td>\n",
       "      <td>299878762615</td>\n",
       "      <td>331104689159</td>\n",
       "      <td>365296539472</td>\n",
       "      <td>402709881799</td>\n",
       "      <td>443620351678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>1</td>\n",
       "      <td>11</td>\n",
       "      <td>58</td>\n",
       "      <td>203</td>\n",
       "      <td>579</td>\n",
       "      <td>1414</td>\n",
       "      <td>3112</td>\n",
       "      <td>6299</td>\n",
       "      <td>11980</td>\n",
       "      <td>21595</td>\n",
       "      <td>...</td>\n",
       "      <td>342875405191</td>\n",
       "      <td>381251130960</td>\n",
       "      <td>423549823086</td>\n",
       "      <td>470135949451</td>\n",
       "      <td>521404745178</td>\n",
       "      <td>577784247998</td>\n",
       "      <td>639738058516</td>\n",
       "      <td>707767673430</td>\n",
       "      <td>782415612693</td>\n",
       "      <td>864268083637</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1</td>\n",
       "      <td>12</td>\n",
       "      <td>62</td>\n",
       "      <td>227</td>\n",
       "      <td>656</td>\n",
       "      <td>1640</td>\n",
       "      <td>3664</td>\n",
       "      <td>7550</td>\n",
       "      <td>14549</td>\n",
       "      <td>26615</td>\n",
       "      <td>...</td>\n",
       "      <td>637106998922</td>\n",
       "      <td>710406879368</td>\n",
       "      <td>791428083525</td>\n",
       "      <td>880912181261</td>\n",
       "      <td>979665072252</td>\n",
       "      <td>1088562592274</td>\n",
       "      <td>1208555392002</td>\n",
       "      <td>1340675335722</td>\n",
       "      <td>1486041108043</td>\n",
       "      <td>1645865507487</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>1</td>\n",
       "      <td>12</td>\n",
       "      <td>67</td>\n",
       "      <td>249</td>\n",
       "      <td>742</td>\n",
       "      <td>1884</td>\n",
       "      <td>4295</td>\n",
       "      <td>8975</td>\n",
       "      <td>17566</td>\n",
       "      <td>32533</td>\n",
       "      <td>...</td>\n",
       "      <td>1160112555500</td>\n",
       "      <td>1297108594549</td>\n",
       "      <td>1448950129971</td>\n",
       "      <td>1617107237728</td>\n",
       "      <td>1803182713077</td>\n",
       "      <td>2008922359789</td>\n",
       "      <td>2236227133256</td>\n",
       "      <td>2487165014643</td>\n",
       "      <td>2763984956628</td>\n",
       "      <td>3069130562462</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>1</td>\n",
       "      <td>13</td>\n",
       "      <td>72</td>\n",
       "      <td>276</td>\n",
       "      <td>833</td>\n",
       "      <td>2164</td>\n",
       "      <td>5000</td>\n",
       "      <td>10624</td>\n",
       "      <td>21060</td>\n",
       "      <td>39533</td>\n",
       "      <td>...</td>\n",
       "      <td>2073252281945</td>\n",
       "      <td>2324203481693</td>\n",
       "      <td>2603086347835</td>\n",
       "      <td>2912748950775</td>\n",
       "      <td>3256303783411</td>\n",
       "      <td>3637151052373</td>\n",
       "      <td>4059001781294</td>\n",
       "      <td>4525904741533</td>\n",
       "      <td>5042273217353</td>\n",
       "      <td>5612916075573</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>1</td>\n",
       "      <td>13</td>\n",
       "      <td>77</td>\n",
       "      <td>301</td>\n",
       "      <td>935</td>\n",
       "      <td>2463</td>\n",
       "      <td>5801</td>\n",
       "      <td>12489</td>\n",
       "      <td>25117</td>\n",
       "      <td>47710</td>\n",
       "      <td>...</td>\n",
       "      <td>3641229062335</td>\n",
       "      <td>4092442040879</td>\n",
       "      <td>4595167656606</td>\n",
       "      <td>5154804303656</td>\n",
       "      <td>5777268190126</td>\n",
       "      <td>6469037559072</td>\n",
       "      <td>7237202379397</td>\n",
       "      <td>8089515858925</td>\n",
       "      <td>9034452164161</td>\n",
       "      <td>10081266305556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>1</td>\n",
       "      <td>14</td>\n",
       "      <td>82</td>\n",
       "      <td>331</td>\n",
       "      <td>1042</td>\n",
       "      <td>2803</td>\n",
       "      <td>6690</td>\n",
       "      <td>14628</td>\n",
       "      <td>29776</td>\n",
       "      <td>57281</td>\n",
       "      <td>...</td>\n",
       "      <td>6292232218259</td>\n",
       "      <td>7089585863616</td>\n",
       "      <td>7980191381884</td>\n",
       "      <td>8974080500569</td>\n",
       "      <td>10082273119093</td>\n",
       "      <td>11316867868605</td>\n",
       "      <td>12691136732040</td>\n",
       "      <td>14219630881129</td>\n",
       "      <td>15918291315366</td>\n",
       "      <td>17804572300436</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>1</td>\n",
       "      <td>14</td>\n",
       "      <td>88</td>\n",
       "      <td>359</td>\n",
       "      <td>1160</td>\n",
       "      <td>3167</td>\n",
       "      <td>7690</td>\n",
       "      <td>17030</td>\n",
       "      <td>35142</td>\n",
       "      <td>68361</td>\n",
       "      <td>...</td>\n",
       "      <td>10709912236632</td>\n",
       "      <td>12096357741240</td>\n",
       "      <td>13648720125966</td>\n",
       "      <td>15385295296395</td>\n",
       "      <td>17326232736733</td>\n",
       "      <td>19493705566858</td>\n",
       "      <td>21912098387841</td>\n",
       "      <td>24608207261659</td>\n",
       "      <td>27611460137231</td>\n",
       "      <td>30954151476020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>1</td>\n",
       "      <td>15</td>\n",
       "      <td>93</td>\n",
       "      <td>393</td>\n",
       "      <td>1285</td>\n",
       "      <td>3575</td>\n",
       "      <td>8796</td>\n",
       "      <td>19764</td>\n",
       "      <td>41254</td>\n",
       "      <td>81220</td>\n",
       "      <td>...</td>\n",
       "      <td>17972508577704</td>\n",
       "      <td>20347114820090</td>\n",
       "      <td>23012193436051</td>\n",
       "      <td>26000542449862</td>\n",
       "      <td>29348365740863</td>\n",
       "      <td>33095600135793</td>\n",
       "      <td>37286265419996</td>\n",
       "      <td>41968850438506</td>\n",
       "      <td>47196726108342</td>\n",
       "      <td>53028600157302</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>1</td>\n",
       "      <td>15</td>\n",
       "      <td>99</td>\n",
       "      <td>424</td>\n",
       "      <td>1422</td>\n",
       "      <td>4011</td>\n",
       "      <td>10031</td>\n",
       "      <td>22816</td>\n",
       "      <td>48239</td>\n",
       "      <td>95993</td>\n",
       "      <td>...</td>\n",
       "      <td>29761177302994</td>\n",
       "      <td>33771015295640</td>\n",
       "      <td>38281754068647</td>\n",
       "      <td>43351240056746</td>\n",
       "      <td>49043478210841</td>\n",
       "      <td>55429231366739</td>\n",
       "      <td>62586679023967</td>\n",
       "      <td>70602128565286</td>\n",
       "      <td>79570795672857</td>\n",
       "      <td>89597646297505</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>1</td>\n",
       "      <td>16</td>\n",
       "      <td>105</td>\n",
       "      <td>461</td>\n",
       "      <td>1566</td>\n",
       "      <td>4499</td>\n",
       "      <td>11389</td>\n",
       "      <td>26263</td>\n",
       "      <td>56149</td>\n",
       "      <td>113000</td>\n",
       "      <td>...</td>\n",
       "      <td>48668952222487</td>\n",
       "      <td>55350536737982</td>\n",
       "      <td>62883715196559</td>\n",
       "      <td>71368980458429</td>\n",
       "      <td>80917770778171</td>\n",
       "      <td>91653571490974</td>\n",
       "      <td>103713109088029</td>\n",
       "      <td>117247663790051</td>\n",
       "      <td>132424491022471</td>\n",
       "      <td>149428381490738</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>71</th>\n",
       "      <td>1</td>\n",
       "      <td>36</td>\n",
       "      <td>491</td>\n",
       "      <td>3917</td>\n",
       "      <td>22400</td>\n",
       "      <td>101488</td>\n",
       "      <td>387562</td>\n",
       "      <td>1296704</td>\n",
       "      <td>3905416</td>\n",
       "      <td>10792075</td>\n",
       "      <td>...</td>\n",
       "      <td>68243097539471307322</td>\n",
       "      <td>82855922431452151895</td>\n",
       "      <td>100444175708971438178</td>\n",
       "      <td>121583493785646279851</td>\n",
       "      <td>146955234777037558538</td>\n",
       "      <td>177364895612857627739</td>\n",
       "      <td>213763569599911266827</td>\n",
       "      <td>257272919664745977803</td>\n",
       "      <td>309214212810216882901</td>\n",
       "      <td>371142041370557573651</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>1</td>\n",
       "      <td>37</td>\n",
       "      <td>504</td>\n",
       "      <td>4068</td>\n",
       "      <td>23451</td>\n",
       "      <td>107176</td>\n",
       "      <td>412264</td>\n",
       "      <td>1388966</td>\n",
       "      <td>4209532</td>\n",
       "      <td>11701171</td>\n",
       "      <td>...</td>\n",
       "      <td>88374125569738949834</td>\n",
       "      <td>107427202509447878305</td>\n",
       "      <td>130387474880785582877</td>\n",
       "      <td>158016452184126056241</td>\n",
       "      <td>191216584570270263262</td>\n",
       "      <td>231056013100043883946</td>\n",
       "      <td>278797440297344966425</td>\n",
       "      <td>335931770697495644028</td>\n",
       "      <td>404217267992489604313</td>\n",
       "      <td>485725086035849614093</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>1</td>\n",
       "      <td>37</td>\n",
       "      <td>517</td>\n",
       "      <td>4213</td>\n",
       "      <td>24544</td>\n",
       "      <td>113061</td>\n",
       "      <td>438260</td>\n",
       "      <td>1486482</td>\n",
       "      <td>4533612</td>\n",
       "      <td>12675060</td>\n",
       "      <td>...</td>\n",
       "      <td>114147883437976609043</td>\n",
       "      <td>138923678192915252177</td>\n",
       "      <td>168815875750330278822</td>\n",
       "      <td>204828943403599639948</td>\n",
       "      <td>248154670139764448025</td>\n",
       "      <td>300205322486120395938</td>\n",
       "      <td>362652365595535712739</td>\n",
       "      <td>437471634624388944439</td>\n",
       "      <td>526995974753799159133</td>\n",
       "      <td>633976520380108981780</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>1</td>\n",
       "      <td>38</td>\n",
       "      <td>530</td>\n",
       "      <td>4371</td>\n",
       "      <td>25668</td>\n",
       "      <td>119244</td>\n",
       "      <td>465539</td>\n",
       "      <td>1589789</td>\n",
       "      <td>4878527</td>\n",
       "      <td>13718392</td>\n",
       "      <td>...</td>\n",
       "      <td>147066386998113768958</td>\n",
       "      <td>179199249695132769539</td>\n",
       "      <td>218013667132307589987</td>\n",
       "      <td>264830814710335511529</td>\n",
       "      <td>321220096518531119074</td>\n",
       "      <td>389043424300406029516</td>\n",
       "      <td>470506989042941971496</td>\n",
       "      <td>568221726911098427733</td>\n",
       "      <td>685273863713656414125</td>\n",
       "      <td>825307130890260352485</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>1</td>\n",
       "      <td>38</td>\n",
       "      <td>544</td>\n",
       "      <td>4523</td>\n",
       "      <td>26835</td>\n",
       "      <td>125642</td>\n",
       "      <td>494211</td>\n",
       "      <td>1698851</td>\n",
       "      <td>5245595</td>\n",
       "      <td>14834467</td>\n",
       "      <td>...</td>\n",
       "      <td>189011107470300078402</td>\n",
       "      <td>230578939834902781229</td>\n",
       "      <td>280849025016319797948</td>\n",
       "      <td>341554284902854378328</td>\n",
       "      <td>414755630343777222849</td>\n",
       "      <td>502900916639217693283</td>\n",
       "      <td>608893954894182049805</td>\n",
       "      <td>736175205435082532308</td>\n",
       "      <td>888816028686005613511</td>\n",
       "      <td>1071628654104984651125</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>1</td>\n",
       "      <td>39</td>\n",
       "      <td>557</td>\n",
       "      <td>4689</td>\n",
       "      <td>28035</td>\n",
       "      <td>132351</td>\n",
       "      <td>524272</td>\n",
       "      <td>1814246</td>\n",
       "      <td>5635739</td>\n",
       "      <td>16028399</td>\n",
       "      <td>...</td>\n",
       "      <td>242333895904094458607</td>\n",
       "      <td>295972476738830886346</td>\n",
       "      <td>360915635127844145719</td>\n",
       "      <td>439430205776750261636</td>\n",
       "      <td>534215178697883558409</td>\n",
       "      <td>648479965110609060777</td>\n",
       "      <td>786036117873315850516</td>\n",
       "      <td>951404700165014299248</td>\n",
       "      <td>1149941835011688574602</td>\n",
       "      <td>1387985356857671262974</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>1</td>\n",
       "      <td>39</td>\n",
       "      <td>571</td>\n",
       "      <td>4848</td>\n",
       "      <td>29280</td>\n",
       "      <td>139289</td>\n",
       "      <td>555832</td>\n",
       "      <td>1935934</td>\n",
       "      <td>6050398</td>\n",
       "      <td>17303816</td>\n",
       "      <td>...</td>\n",
       "      <td>309968559679379116946</td>\n",
       "      <td>379013839454881552037</td>\n",
       "      <td>462706891553900128714</td>\n",
       "      <td>564005119249391328660</td>\n",
       "      <td>686433761659870977746</td>\n",
       "      <td>834189496817675264681</td>\n",
       "      <td>1012261992817391987857</td>\n",
       "      <td>1226576356921400691567</td>\n",
       "      <td>1484159892189237772176</td>\n",
       "      <td>1793337098558786453476</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>1</td>\n",
       "      <td>40</td>\n",
       "      <td>585</td>\n",
       "      <td>5021</td>\n",
       "      <td>30559</td>\n",
       "      <td>146561</td>\n",
       "      <td>588886</td>\n",
       "      <td>2064528</td>\n",
       "      <td>6490594</td>\n",
       "      <td>18666305</td>\n",
       "      <td>...</td>\n",
       "      <td>395567534582575885535</td>\n",
       "      <td>484232389740932479030</td>\n",
       "      <td>591829773075788318944</td>\n",
       "      <td>722208061104937649429</td>\n",
       "      <td>879959738280195860086</td>\n",
       "      <td>1070558155390840039741</td>\n",
       "      <td>1300518160307331586078</td>\n",
       "      <td>1577584555845720504762</td>\n",
       "      <td>1910952960709582784235</td>\n",
       "      <td>2311528363295492026655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>1</td>\n",
       "      <td>40</td>\n",
       "      <td>599</td>\n",
       "      <td>5188</td>\n",
       "      <td>31885</td>\n",
       "      <td>154071</td>\n",
       "      <td>623556</td>\n",
       "      <td>2199997</td>\n",
       "      <td>6957847</td>\n",
       "      <td>20119883</td>\n",
       "      <td>...</td>\n",
       "      <td>503669001156752919235</td>\n",
       "      <td>617262366536480647180</td>\n",
       "      <td>755266965287006035195</td>\n",
       "      <td>922677922876700716224</td>\n",
       "      <td>1125462901762382515638</td>\n",
       "      <td>1370742131374338202413</td>\n",
       "      <td>1667000099675889765300</td>\n",
       "      <td>2024334192313292206910</td>\n",
       "      <td>2454746402619040315870</td>\n",
       "      <td>2972485198440800007878</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>1</td>\n",
       "      <td>41</td>\n",
       "      <td>613</td>\n",
       "      <td>5369</td>\n",
       "      <td>33247</td>\n",
       "      <td>161934</td>\n",
       "      <td>659832</td>\n",
       "      <td>2342988</td>\n",
       "      <td>7453287</td>\n",
       "      <td>21670632</td>\n",
       "      <td>...</td>\n",
       "      <td>639900873524254659271</td>\n",
       "      <td>785098896802920675821</td>\n",
       "      <td>961697549516433155815</td>\n",
       "      <td>1176164410248392110264</td>\n",
       "      <td>1436234882455524504028</td>\n",
       "      <td>1751148553708423535575</td>\n",
       "      <td>2131927430112364849713</td>\n",
       "      <td>2591703090982426842590</td>\n",
       "      <td>3146100933381274347846</td>\n",
       "      <td>3813690971111761213960</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>1</td>\n",
       "      <td>41</td>\n",
       "      <td>628</td>\n",
       "      <td>5543</td>\n",
       "      <td>34657</td>\n",
       "      <td>170056</td>\n",
       "      <td>697841</td>\n",
       "      <td>2493466</td>\n",
       "      <td>7978570</td>\n",
       "      <td>23322966</td>\n",
       "      <td>...</td>\n",
       "      <td>811229374684006718093</td>\n",
       "      <td>996410318411272648553</td>\n",
       "      <td>1221888672241632106771</td>\n",
       "      <td>1496018297479714763544</td>\n",
       "      <td>1828801674668586731631</td>\n",
       "      <td>2232199405524590371943</td>\n",
       "      <td>2720494947726337193097</td>\n",
       "      <td>3310723945128516690604</td>\n",
       "      <td>4023179022478493872778</td>\n",
       "      <td>4882002649836878450104</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>1</td>\n",
       "      <td>42</td>\n",
       "      <td>642</td>\n",
       "      <td>5732</td>\n",
       "      <td>36105</td>\n",
       "      <td>178545</td>\n",
       "      <td>737580</td>\n",
       "      <td>2652127</td>\n",
       "      <td>8534888</td>\n",
       "      <td>25083536</td>\n",
       "      <td>...</td>\n",
       "      <td>1026261441437418107018</td>\n",
       "      <td>1261918566008448121235</td>\n",
       "      <td>1549173103070339455921</td>\n",
       "      <td>1898789853229383079263</td>\n",
       "      <td>2323672138109491699050</td>\n",
       "      <td>2839266043565498164860</td>\n",
       "      <td>3464037320280788488866</td>\n",
       "      <td>4220033349950460841865</td>\n",
       "      <td>5133544596983895280775</td>\n",
       "      <td>6235882287161873662121</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>1</td>\n",
       "      <td>42</td>\n",
       "      <td>657</td>\n",
       "      <td>5914</td>\n",
       "      <td>37603</td>\n",
       "      <td>187308</td>\n",
       "      <td>779175</td>\n",
       "      <td>2818936</td>\n",
       "      <td>9124039</td>\n",
       "      <td>26957168</td>\n",
       "      <td>...</td>\n",
       "      <td>1295612018442193294465</td>\n",
       "      <td>1594861695556547444622</td>\n",
       "      <td>1960030558823320268670</td>\n",
       "      <td>2404958097664731572568</td>\n",
       "      <td>2946251138516544996472</td>\n",
       "      <td>3603810510560083814107</td>\n",
       "      <td>4401453041747464879628</td>\n",
       "      <td>5367645284748487528525</td>\n",
       "      <td>6536368051974237347663</td>\n",
       "      <td>7948133933474026773782</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>1</td>\n",
       "      <td>43</td>\n",
       "      <td>672</td>\n",
       "      <td>6111</td>\n",
       "      <td>39140</td>\n",
       "      <td>196464</td>\n",
       "      <td>822624</td>\n",
       "      <td>2994625</td>\n",
       "      <td>9747338</td>\n",
       "      <td>28951095</td>\n",
       "      <td>...</td>\n",
       "      <td>1632349455169638573160</td>\n",
       "      <td>2011555383957819756449</td>\n",
       "      <td>2474794201062602626934</td>\n",
       "      <td>3039818054802651670244</td>\n",
       "      <td>3727951726805334556326</td>\n",
       "      <td>4564777117304652911147</td>\n",
       "      <td>5580942490200377989218</td>\n",
       "      <td>6813118095793868754831</td>\n",
       "      <td>8305123347426919996570</td>\n",
       "      <td>10109254842978271333700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>1</td>\n",
       "      <td>43</td>\n",
       "      <td>687</td>\n",
       "      <td>6301</td>\n",
       "      <td>40730</td>\n",
       "      <td>205905</td>\n",
       "      <td>868065</td>\n",
       "      <td>3179167</td>\n",
       "      <td>10406681</td>\n",
       "      <td>31070634</td>\n",
       "      <td>...</td>\n",
       "      <td>2052534771609447212864</td>\n",
       "      <td>2532073513880073019681</td>\n",
       "      <td>3118507905149903785104</td>\n",
       "      <td>3834558516763311728987</td>\n",
       "      <td>4707547579293465770851</td>\n",
       "      <td>5770286450515883198460</td>\n",
       "      <td>7062125951402879380625</td>\n",
       "      <td>8630197992440904463294</td>\n",
       "      <td>10530881626019872615142</td>\n",
       "      <td>12831532115865869261911</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>1</td>\n",
       "      <td>44</td>\n",
       "      <td>702</td>\n",
       "      <td>6506</td>\n",
       "      <td>42360</td>\n",
       "      <td>215760</td>\n",
       "      <td>915491</td>\n",
       "      <td>3373336</td>\n",
       "      <td>11103509</td>\n",
       "      <td>33323620</td>\n",
       "      <td>...</td>\n",
       "      <td>2575873577593806450404</td>\n",
       "      <td>3181071831295384594585</td>\n",
       "      <td>3921964867365622890869</td>\n",
       "      <td>4827569190102142440751</td>\n",
       "      <td>5932815028550450580465</td>\n",
       "      <td>7279694286431712038773</td>\n",
       "      <td>8918620590291610345692</td>\n",
       "      <td>10910038709956984032601</td>\n",
       "      <td>13326326851436911697969</td>\n",
       "      <td>16254042536775417511870</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>1</td>\n",
       "      <td>44</td>\n",
       "      <td>718</td>\n",
       "      <td>6704</td>\n",
       "      <td>44044</td>\n",
       "      <td>225922</td>\n",
       "      <td>965045</td>\n",
       "      <td>3577104</td>\n",
       "      <td>11839876</td>\n",
       "      <td>35715866</td>\n",
       "      <td>...</td>\n",
       "      <td>3226502997959586172190</td>\n",
       "      <td>3988783249333951077225</td>\n",
       "      <td>4922964000071241498024</td>\n",
       "      <td>6066023625389531975382</td>\n",
       "      <td>7462523634805515147285</td>\n",
       "      <td>9166090153477392630696</td>\n",
       "      <td>11241170954234462115338</td>\n",
       "      <td>13765116801041977223356</td>\n",
       "      <td>16830644068586804597558</td>\n",
       "      <td>20548745117449190991672</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>1</td>\n",
       "      <td>45</td>\n",
       "      <td>733</td>\n",
       "      <td>6918</td>\n",
       "      <td>45771</td>\n",
       "      <td>236514</td>\n",
       "      <td>1016727</td>\n",
       "      <td>3791297</td>\n",
       "      <td>12617305</td>\n",
       "      <td>38255902</td>\n",
       "      <td>...</td>\n",
       "      <td>4033940165302745868620</td>\n",
       "      <td>4992218790126889765403</td>\n",
       "      <td>6167827522505794400432</td>\n",
       "      <td>7607793247315951559554</td>\n",
       "      <td>9368845648861630281158</td>\n",
       "      <td>11519324836522283552127</td>\n",
       "      <td>14141446120049026513095</td>\n",
       "      <td>17333985567314390480713</td>\n",
       "      <td>21215460758848880865748</td>\n",
       "      <td>25927893671751287642590</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>1</td>\n",
       "      <td>45</td>\n",
       "      <td>749</td>\n",
       "      <td>7124</td>\n",
       "      <td>47554</td>\n",
       "      <td>247431</td>\n",
       "      <td>1070681</td>\n",
       "      <td>4015884</td>\n",
       "      <td>13438017</td>\n",
       "      <td>40950056</td>\n",
       "      <td>...</td>\n",
       "      <td>5034223803901998840762</td>\n",
       "      <td>6236614548846586327294</td>\n",
       "      <td>7713231373185120322973</td>\n",
       "      <td>9523758344697041448120</td>\n",
       "      <td>11740268212583524838457</td>\n",
       "      <td>14449673353755444902016</td>\n",
       "      <td>17756638573307399970930</td>\n",
       "      <td>21787038601763603644884</td>\n",
       "      <td>26692057266971247975858</td>\n",
       "      <td>32653041745836311514237</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>1</td>\n",
       "      <td>46</td>\n",
       "      <td>765</td>\n",
       "      <td>7346</td>\n",
       "      <td>49381</td>\n",
       "      <td>258806</td>\n",
       "      <td>1126907</td>\n",
       "      <td>4251736</td>\n",
       "      <td>14303676</td>\n",
       "      <td>43807567</td>\n",
       "      <td>...</td>\n",
       "      <td>6271286269342473176317</td>\n",
       "      <td>7777172598004191915040</td>\n",
       "      <td>9628409764968851285892</td>\n",
       "      <td>11900594329971417027321</td>\n",
       "      <td>14685108096348533100878</td>\n",
       "      <td>18092260337835667031747</td>\n",
       "      <td>22255025941669814291789</td>\n",
       "      <td>27333487068925226686742</td>\n",
       "      <td>33520104418774811690603</td>\n",
       "      <td>41045965689724323609166</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>1</td>\n",
       "      <td>46</td>\n",
       "      <td>781</td>\n",
       "      <td>7561</td>\n",
       "      <td>51266</td>\n",
       "      <td>270518</td>\n",
       "      <td>1185562</td>\n",
       "      <td>4498835</td>\n",
       "      <td>15216621</td>\n",
       "      <td>46835360</td>\n",
       "      <td>...</td>\n",
       "      <td>7798600276821377194985</td>\n",
       "      <td>9681152617752581022626</td>\n",
       "      <td>11997806628232945434683</td>\n",
       "      <td>14844126262949498343640</td>\n",
       "      <td>18335747610404248124160</td>\n",
       "      <td>22612398865312835945661</td>\n",
       "      <td>27842687506244864383096</td>\n",
       "      <td>34229794117204917979805</td>\n",
       "      <td>42018236411383147341983</td>\n",
       "      <td>51501895102005980475789</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>1</td>\n",
       "      <td>47</td>\n",
       "      <td>797</td>\n",
       "      <td>7792</td>\n",
       "      <td>53197</td>\n",
       "      <td>282711</td>\n",
       "      <td>1246640</td>\n",
       "      <td>4758095</td>\n",
       "      <td>16178669</td>\n",
       "      <td>50043415</td>\n",
       "      <td>...</td>\n",
       "      <td>9681152617752581022626</td>\n",
       "      <td>12030381464338466788530</td>\n",
       "      <td>14924260133016185345961</td>\n",
       "      <td>18483361935520255483718</td>\n",
       "      <td>22853732542820411718525</td>\n",
       "      <td>28212022244946197365013</td>\n",
       "      <td>34771603814764673033251</td>\n",
       "      <td>42789856269351189071663</td>\n",
       "      <td>52576826217674483587913</td>\n",
       "      <td>64505515080297080498839</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>1</td>\n",
       "      <td>47</td>\n",
       "      <td>814</td>\n",
       "      <td>8015</td>\n",
       "      <td>55188</td>\n",
       "      <td>295266</td>\n",
       "      <td>1310304</td>\n",
       "      <td>5029495</td>\n",
       "      <td>17192343</td>\n",
       "      <td>53439272</td>\n",
       "      <td>...</td>\n",
       "      <td>11997806628232945434683</td>\n",
       "      <td>14924260133016185345961</td>\n",
       "      <td>18532822287896171259214</td>\n",
       "      <td>22975334174330703957244</td>\n",
       "      <td>28435899147241611876444</td>\n",
       "      <td>35137421841992200284574</td>\n",
       "      <td>43349410685715738092838</td>\n",
       "      <td>53397276519895299681866</td>\n",
       "      <td>65673400715356106371121</td>\n",
       "      <td>80650293735239459932684</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>1</td>\n",
       "      <td>48</td>\n",
       "      <td>830</td>\n",
       "      <td>8255</td>\n",
       "      <td>57227</td>\n",
       "      <td>308320</td>\n",
       "      <td>1376557</td>\n",
       "      <td>5314010</td>\n",
       "      <td>18259557</td>\n",
       "      <td>57033742</td>\n",
       "      <td>...</td>\n",
       "      <td>14844126262949498343640</td>\n",
       "      <td>18483361935520255483718</td>\n",
       "      <td>22975334174330703957244</td>\n",
       "      <td>28510906951787097523423</td>\n",
       "      <td>35321727996295542352946</td>\n",
       "      <td>43688543551438043098422</td>\n",
       "      <td>53951129547247609546474</td>\n",
       "      <td>66520138639050296645877</td>\n",
       "      <td>81891215404061498831148</td>\n",
       "      <td>100661793694507454764302</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>1</td>\n",
       "      <td>48</td>\n",
       "      <td>847</td>\n",
       "      <td>8487</td>\n",
       "      <td>59328</td>\n",
       "      <td>321755</td>\n",
       "      <td>1445561</td>\n",
       "      <td>5611621</td>\n",
       "      <td>19383028</td>\n",
       "      <td>60835001</td>\n",
       "      <td>...</td>\n",
       "      <td>18335747610404248124160</td>\n",
       "      <td>22853732542820411718525</td>\n",
       "      <td>28435899147241611876444</td>\n",
       "      <td>35321727996295542352946</td>\n",
       "      <td>43802158708484824598491</td>\n",
       "      <td>54230142045709761859817</td>\n",
       "      <td>67033255708747064116389</td>\n",
       "      <td>82728768532796604957656</td>\n",
       "      <td>101941606100399065396598</td>\n",
       "      <td>125425750652159548322803</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>1</td>\n",
       "      <td>49</td>\n",
       "      <td>864</td>\n",
       "      <td>8736</td>\n",
       "      <td>61478</td>\n",
       "      <td>335720</td>\n",
       "      <td>1517320</td>\n",
       "      <td>5923350</td>\n",
       "      <td>20564840</td>\n",
       "      <td>64854717</td>\n",
       "      <td>...</td>\n",
       "      <td>22612398865312835945661</td>\n",
       "      <td>28212022244946197365013</td>\n",
       "      <td>35137421841992200284574</td>\n",
       "      <td>43688543551438043098422</td>\n",
       "      <td>54230142045709761859817</td>\n",
       "      <td>67205146583816033597718</td>\n",
       "      <td>83150656334510766860102</td>\n",
       "      <td>102717058409866322970452</td>\n",
       "      <td>126690849944416153037449</td>\n",
       "      <td>156021848694303105845536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>1</td>\n",
       "      <td>49</td>\n",
       "      <td>881</td>\n",
       "      <td>8977</td>\n",
       "      <td>63693</td>\n",
       "      <td>350080</td>\n",
       "      <td>1592010</td>\n",
       "      <td>6249188</td>\n",
       "      <td>21807851</td>\n",
       "      <td>69101801</td>\n",
       "      <td>...</td>\n",
       "      <td>27842687506244864383096</td>\n",
       "      <td>34771603814764673033251</td>\n",
       "      <td>43349410685715738092838</td>\n",
       "      <td>53951129547247609546474</td>\n",
       "      <td>67033255708747064116389</td>\n",
       "      <td>83150656334510766860102</td>\n",
       "      <td>102976812343877587172799</td>\n",
       "      <td>127328035243255436873229</td>\n",
       "      <td>157192403373937340065436</td>\n",
       "      <td>193764294973792840469175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>1</td>\n",
       "      <td>50</td>\n",
       "      <td>898</td>\n",
       "      <td>9235</td>\n",
       "      <td>65959</td>\n",
       "      <td>364995</td>\n",
       "      <td>1669629</td>\n",
       "      <td>6590210</td>\n",
       "      <td>23114319</td>\n",
       "      <td>73588810</td>\n",
       "      <td>...</td>\n",
       "      <td>34229794117204917979805</td>\n",
       "      <td>42789856269351189071663</td>\n",
       "      <td>53397276519895299681866</td>\n",
       "      <td>66520138639050296645877</td>\n",
       "      <td>82728768532796604957656</td>\n",
       "      <td>102717058409866322970452</td>\n",
       "      <td>127328035243255436873229</td>\n",
       "      <td>157584479243809410583714</td>\n",
       "      <td>194725543861235161461969</td>\n",
       "      <td>240250499761577776466542</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>1</td>\n",
       "      <td>50</td>\n",
       "      <td>916</td>\n",
       "      <td>9485</td>\n",
       "      <td>68291</td>\n",
       "      <td>380332</td>\n",
       "      <td>1750359</td>\n",
       "      <td>6946409</td>\n",
       "      <td>24487316</td>\n",
       "      <td>78325400</td>\n",
       "      <td>...</td>\n",
       "      <td>42018236411383147341983</td>\n",
       "      <td>52576826217674483587913</td>\n",
       "      <td>65673400715356106371121</td>\n",
       "      <td>81891215404061498831148</td>\n",
       "      <td>101941606100399065396598</td>\n",
       "      <td>126690849944416153037449</td>\n",
       "      <td>157192403373937340065436</td>\n",
       "      <td>194725543861235161461969</td>\n",
       "      <td>240841629152665669312110</td>\n",
       "      <td>297419407128909181291513</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>1</td>\n",
       "      <td>51</td>\n",
       "      <td>933</td>\n",
       "      <td>9753</td>\n",
       "      <td>70677</td>\n",
       "      <td>396244</td>\n",
       "      <td>1834207</td>\n",
       "      <td>7318923</td>\n",
       "      <td>25929222</td>\n",
       "      <td>83325131</td>\n",
       "      <td>...</td>\n",
       "      <td>51501895102005980475789</td>\n",
       "      <td>64505515080297080498839</td>\n",
       "      <td>80650293735239459932684</td>\n",
       "      <td>100661793694507454764302</td>\n",
       "      <td>125425750652159548322803</td>\n",
       "      <td>156021848694303105845536</td>\n",
       "      <td>193764294973792840469175</td>\n",
       "      <td>240250499761577776466542</td>\n",
       "      <td>297419407128909181291513</td>\n",
       "      <td>367621302506997599598267</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 100 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "     1    2    3     4      5       6        7        8         9         10   \\\n",
       "1      1    1    1     1      1       1        1        1         1         1   \n",
       "2      1    2    2     3      3       4        4        5         5         6   \n",
       "3      1    2    4     5      7       9       11       13        16        18   \n",
       "4      1    3    5     9     12      17       22       29        35        44   \n",
       "5      1    3    7    12     20      28       40       53        70        89   \n",
       "6      1    4    9    17     28      45       64       91       123       164   \n",
       "7      1    4   11    22     40      64      100      144       204       278   \n",
       "8      1    5   13    29     53      91      144      220       317       449   \n",
       "9      1    5   16    35     70     123      204      317       478       688   \n",
       "10     1    6   18    44     89     164      278      449       688      1025   \n",
       "11     1    6   21    52    112     211      373      614       971      1471   \n",
       "12     1    7   24    63    137     271      486      827      1332      2069   \n",
       "13     1    7   27    73    168     337      628     1087      1797      2840   \n",
       "14     1    8   30    86    201     419      794     1413      2375      3837   \n",
       "15     1    8   34    98    240     510      995     1802      3101      5087   \n",
       "16     1    9   37   114    282     619     1229     2280      3983      6664   \n",
       "17     1    9   41   128    331     739     1507     2841      5067      8602   \n",
       "18     1   10   45   146    383     882     1825     3516      6366     10995   \n",
       "19     1   10   49   163    443    1036     2199     4301      7930     13890   \n",
       "20     1   11   53   184    507    1217     2622     5231      9780     17404   \n",
       "21     1   11   58   203    579    1414     3112     6299     11980     21595   \n",
       "22     1   12   62   227    656    1640     3664     7550     14549     26615   \n",
       "23     1   12   67   249    742    1884     4295     8975     17566     32533   \n",
       "24     1   13   72   276    833    2164     5000    10624     21060     39533   \n",
       "25     1   13   77   301    935    2463     5801    12489     25117     47710   \n",
       "26     1   14   82   331   1042    2803     6690    14628     29776     57281   \n",
       "27     1   14   88   359   1160    3167     7690    17030     35142     68361   \n",
       "28     1   15   93   393   1285    3575     8796    19764     41254     81220   \n",
       "29     1   15   99   424   1422    4011    10031    22816     48239     95993   \n",
       "30     1   16  105   461   1566    4499    11389    26263     56149    113000   \n",
       "..   ...  ...  ...   ...    ...     ...      ...      ...       ...       ...   \n",
       "71     1   36  491  3917  22400  101488   387562  1296704   3905416  10792075   \n",
       "72     1   37  504  4068  23451  107176   412264  1388966   4209532  11701171   \n",
       "73     1   37  517  4213  24544  113061   438260  1486482   4533612  12675060   \n",
       "74     1   38  530  4371  25668  119244   465539  1589789   4878527  13718392   \n",
       "75     1   38  544  4523  26835  125642   494211  1698851   5245595  14834467   \n",
       "76     1   39  557  4689  28035  132351   524272  1814246   5635739  16028399   \n",
       "77     1   39  571  4848  29280  139289   555832  1935934   6050398  17303816   \n",
       "78     1   40  585  5021  30559  146561   588886  2064528   6490594  18666305   \n",
       "79     1   40  599  5188  31885  154071   623556  2199997   6957847  20119883   \n",
       "80     1   41  613  5369  33247  161934   659832  2342988   7453287  21670632   \n",
       "81     1   41  628  5543  34657  170056   697841  2493466   7978570  23322966   \n",
       "82     1   42  642  5732  36105  178545   737580  2652127   8534888  25083536   \n",
       "83     1   42  657  5914  37603  187308   779175  2818936   9124039  26957168   \n",
       "84     1   43  672  6111  39140  196464   822624  2994625   9747338  28951095   \n",
       "85     1   43  687  6301  40730  205905   868065  3179167  10406681  31070634   \n",
       "86     1   44  702  6506  42360  215760   915491  3373336  11103509  33323620   \n",
       "87     1   44  718  6704  44044  225922   965045  3577104  11839876  35715866   \n",
       "88     1   45  733  6918  45771  236514  1016727  3791297  12617305  38255902   \n",
       "89     1   45  749  7124  47554  247431  1070681  4015884  13438017  40950056   \n",
       "90     1   46  765  7346  49381  258806  1126907  4251736  14303676  43807567   \n",
       "91     1   46  781  7561  51266  270518  1185562  4498835  15216621  46835360   \n",
       "92     1   47  797  7792  53197  282711  1246640  4758095  16178669  50043415   \n",
       "93     1   47  814  8015  55188  295266  1310304  5029495  17192343  53439272   \n",
       "94     1   48  830  8255  57227  308320  1376557  5314010  18259557  57033742   \n",
       "95     1   48  847  8487  59328  321755  1445561  5611621  19383028  60835001   \n",
       "96     1   49  864  8736  61478  335720  1517320  5923350  20564840  64854717   \n",
       "97     1   49  881  8977  63693  350080  1592010  6249188  21807851  69101801   \n",
       "98     1   50  898  9235  65959  364995  1669629  6590210  23114319  73588810   \n",
       "99     1   50  916  9485  68291  380332  1750359  6946409  24487316  78325400   \n",
       "100    1   51  933  9753  70677  396244  1834207  7318923  25929222  83325131   \n",
       "\n",
       "               ...                                 91   \\\n",
       "1              ...                                   1   \n",
       "2              ...                                  46   \n",
       "3              ...                                 781   \n",
       "4              ...                                7561   \n",
       "5              ...                               51266   \n",
       "6              ...                              270518   \n",
       "7              ...                             1185562   \n",
       "8              ...                             4498835   \n",
       "9              ...                            15216621   \n",
       "10             ...                            46835360   \n",
       "11             ...                           133229118   \n",
       "12             ...                           354430164   \n",
       "13             ...                           890030320   \n",
       "14             ...                          2125398724   \n",
       "15             ...                          4855757574   \n",
       "16             ...                         10666233255   \n",
       "17             ...                         22620607479   \n",
       "18             ...                         46478894526   \n",
       "19             ...                         92802791172   \n",
       "20             ...                        180523195041   \n",
       "21             ...                        342875405191   \n",
       "22             ...                        637106998922   \n",
       "23             ...                       1160112555500   \n",
       "24             ...                       2073252281945   \n",
       "25             ...                       3641229062335   \n",
       "26             ...                       6292232218259   \n",
       "27             ...                      10709912236632   \n",
       "28             ...                      17972508577704   \n",
       "29             ...                      29761177302994   \n",
       "30             ...                      48668952222487   \n",
       "..             ...                                 ...   \n",
       "71             ...                68243097539471307322   \n",
       "72             ...                88374125569738949834   \n",
       "73             ...               114147883437976609043   \n",
       "74             ...               147066386998113768958   \n",
       "75             ...               189011107470300078402   \n",
       "76             ...               242333895904094458607   \n",
       "77             ...               309968559679379116946   \n",
       "78             ...               395567534582575885535   \n",
       "79             ...               503669001156752919235   \n",
       "80             ...               639900873524254659271   \n",
       "81             ...               811229374684006718093   \n",
       "82             ...              1026261441437418107018   \n",
       "83             ...              1295612018442193294465   \n",
       "84             ...              1632349455169638573160   \n",
       "85             ...              2052534771609447212864   \n",
       "86             ...              2575873577593806450404   \n",
       "87             ...              3226502997959586172190   \n",
       "88             ...              4033940165302745868620   \n",
       "89             ...              5034223803901998840762   \n",
       "90             ...              6271286269342473176317   \n",
       "91             ...              7798600276821377194985   \n",
       "92             ...              9681152617752581022626   \n",
       "93             ...             11997806628232945434683   \n",
       "94             ...             14844126262949498343640   \n",
       "95             ...             18335747610404248124160   \n",
       "96             ...             22612398865312835945661   \n",
       "97             ...             27842687506244864383096   \n",
       "98             ...             34229794117204917979805   \n",
       "99             ...             42018236411383147341983   \n",
       "100            ...             51501895102005980475789   \n",
       "\n",
       "                         92                       93   \\\n",
       "1                          1                        1   \n",
       "2                         47                       47   \n",
       "3                        797                      814   \n",
       "4                       7792                     8015   \n",
       "5                      53197                    55188   \n",
       "6                     282711                   295266   \n",
       "7                    1246640                  1310304   \n",
       "8                    4758095                  5029495   \n",
       "9                   16178669                 17192343   \n",
       "10                  50043415                 53439272   \n",
       "11                 143012122                153419266   \n",
       "12                 382112616                411687438   \n",
       "13                 963484803               1042293366   \n",
       "14                2309777794               2508366675   \n",
       "15                5296557634               5773120842   \n",
       "16               11675695467              12770935482   \n",
       "17               24845334433              27267375662   \n",
       "18               51216088034              56390280046   \n",
       "19              102581140543             113295193417   \n",
       "20              200145723313             221711050361   \n",
       "21              381251130960             423549823086   \n",
       "22              710406879368             791428083525   \n",
       "23             1297108594549            1448950129971   \n",
       "24             2324203481693            2603086347835   \n",
       "25             4092442040879            4595167656606   \n",
       "26             7089585863616            7980191381884   \n",
       "27            12096357741240           13648720125966   \n",
       "28            20347114820090           23012193436051   \n",
       "29            33771015295640           38281754068647   \n",
       "30            55350536737982           62883715196559   \n",
       "..                       ...                      ...   \n",
       "71      82855922431452151895    100444175708971438178   \n",
       "72     107427202509447878305    130387474880785582877   \n",
       "73     138923678192915252177    168815875750330278822   \n",
       "74     179199249695132769539    218013667132307589987   \n",
       "75     230578939834902781229    280849025016319797948   \n",
       "76     295972476738830886346    360915635127844145719   \n",
       "77     379013839454881552037    462706891553900128714   \n",
       "78     484232389740932479030    591829773075788318944   \n",
       "79     617262366536480647180    755266965287006035195   \n",
       "80     785098896802920675821    961697549516433155815   \n",
       "81     996410318411272648553   1221888672241632106771   \n",
       "82    1261918566008448121235   1549173103070339455921   \n",
       "83    1594861695556547444622   1960030558823320268670   \n",
       "84    2011555383957819756449   2474794201062602626934   \n",
       "85    2532073513880073019681   3118507905149903785104   \n",
       "86    3181071831295384594585   3921964867365622890869   \n",
       "87    3988783249333951077225   4922964000071241498024   \n",
       "88    4992218790126889765403   6167827522505794400432   \n",
       "89    6236614548846586327294   7713231373185120322973   \n",
       "90    7777172598004191915040   9628409764968851285892   \n",
       "91    9681152617752581022626  11997806628232945434683   \n",
       "92   12030381464338466788530  14924260133016185345961   \n",
       "93   14924260133016185345961  18532822287896171259214   \n",
       "94   18483361935520255483718  22975334174330703957244   \n",
       "95   22853732542820411718525  28435899147241611876444   \n",
       "96   28212022244946197365013  35137421841992200284574   \n",
       "97   34771603814764673033251  43349410685715738092838   \n",
       "98   42789856269351189071663  53397276519895299681866   \n",
       "99   52576826217674483587913  65673400715356106371121   \n",
       "100  64505515080297080498839  80650293735239459932684   \n",
       "\n",
       "                          94                        95   \\\n",
       "1                           1                         1   \n",
       "2                          48                        48   \n",
       "3                         830                       847   \n",
       "4                        8255                      8487   \n",
       "5                       57227                     59328   \n",
       "6                      308320                    321755   \n",
       "7                     1376557                   1445561   \n",
       "8                     5314010                   5611621   \n",
       "9                    18259557                  19383028   \n",
       "10                   57033742                  60835001   \n",
       "11                  164483447                 176241298   \n",
       "12                  443270455                 476975561   \n",
       "13                 1126793603                1217347814   \n",
       "14                 2722138571                2952105831   \n",
       "15                 6288010239                6843966154   \n",
       "16                13958469068               15245208885   \n",
       "17                29902387794               32767167110   \n",
       "18                62037774735               68197490525   \n",
       "19               125025773840              137860045355   \n",
       "20               245393563798              271381601376   \n",
       "21               470135949451              521404745178   \n",
       "22               880912181261              979665072252   \n",
       "23              1617107237728             1803182713077   \n",
       "24              2912748950775             3256303783411   \n",
       "25              5154804303656             5777268190126   \n",
       "26              8974080500569            10082273119093   \n",
       "27             15385295296395            17326232736733   \n",
       "28             26000542449862            29348365740863   \n",
       "29             43351240056746            49043478210841   \n",
       "30             71368980458429            80917770778171   \n",
       "..                        ...                       ...   \n",
       "71      121583493785646279851     146955234777037558538   \n",
       "72      158016452184126056241     191216584570270263262   \n",
       "73      204828943403599639948     248154670139764448025   \n",
       "74      264830814710335511529     321220096518531119074   \n",
       "75      341554284902854378328     414755630343777222849   \n",
       "76      439430205776750261636     534215178697883558409   \n",
       "77      564005119249391328660     686433761659870977746   \n",
       "78      722208061104937649429     879959738280195860086   \n",
       "79      922677922876700716224    1125462901762382515638   \n",
       "80     1176164410248392110264    1436234882455524504028   \n",
       "81     1496018297479714763544    1828801674668586731631   \n",
       "82     1898789853229383079263    2323672138109491699050   \n",
       "83     2404958097664731572568    2946251138516544996472   \n",
       "84     3039818054802651670244    3727951726805334556326   \n",
       "85     3834558516763311728987    4707547579293465770851   \n",
       "86     4827569190102142440751    5932815028550450580465   \n",
       "87     6066023625389531975382    7462523634805515147285   \n",
       "88     7607793247315951559554    9368845648861630281158   \n",
       "89     9523758344697041448120   11740268212583524838457   \n",
       "90    11900594329971417027321   14685108096348533100878   \n",
       "91    14844126262949498343640   18335747610404248124160   \n",
       "92    18483361935520255483718   22853732542820411718525   \n",
       "93    22975334174330703957244   28435899147241611876444   \n",
       "94    28510906951787097523423   35321727996295542352946   \n",
       "95    35321727996295542352946   43802158708484824598491   \n",
       "96    43688543551438043098422   54230142045709761859817   \n",
       "97    53951129547247609546474   67033255708747064116389   \n",
       "98    66520138639050296645877   82728768532796604957656   \n",
       "99    81891215404061498831148  101941606100399065396598   \n",
       "100  100661793694507454764302  125425750652159548322803   \n",
       "\n",
       "                          96                        97   \\\n",
       "1                           1                         1   \n",
       "2                          49                        49   \n",
       "3                         864                       881   \n",
       "4                        8736                      8977   \n",
       "5                       61478                     63693   \n",
       "6                      335720                    350080   \n",
       "7                     1517320                   1592010   \n",
       "8                     5923350                   6249188   \n",
       "9                    20564840                  21807851   \n",
       "10                   64854717                  69101801   \n",
       "11                  188728799                 201985954   \n",
       "12                  512930426                 551260571   \n",
       "13                 1314331599                1418147515   \n",
       "14                 3199359087                3465032741   \n",
       "15                 7443884292                8090856556   \n",
       "16                16638567213               18146402339   \n",
       "17                35879650577               39259059190   \n",
       "18                74911289793               82224005011   \n",
       "19               151891803561              167222083347   \n",
       "20               299878762615              331104689159   \n",
       "21               577784247998              639738058516   \n",
       "22              1088562592274             1208555392002   \n",
       "23              2008922359789             2236227133256   \n",
       "24              3637151052373             4059001781294   \n",
       "25              6469037559072             7237202379397   \n",
       "26             11316867868605            12691136732040   \n",
       "27             19493705566858            21912098387841   \n",
       "28             33095600135793            37286265419996   \n",
       "29             55429231366739            62586679023967   \n",
       "30             91653571490974           103713109088029   \n",
       "..                        ...                       ...   \n",
       "71      177364895612857627739     213763569599911266827   \n",
       "72      231056013100043883946     278797440297344966425   \n",
       "73      300205322486120395938     362652365595535712739   \n",
       "74      389043424300406029516     470506989042941971496   \n",
       "75      502900916639217693283     608893954894182049805   \n",
       "76      648479965110609060777     786036117873315850516   \n",
       "77      834189496817675264681    1012261992817391987857   \n",
       "78     1070558155390840039741    1300518160307331586078   \n",
       "79     1370742131374338202413    1667000099675889765300   \n",
       "80     1751148553708423535575    2131927430112364849713   \n",
       "81     2232199405524590371943    2720494947726337193097   \n",
       "82     2839266043565498164860    3464037320280788488866   \n",
       "83     3603810510560083814107    4401453041747464879628   \n",
       "84     4564777117304652911147    5580942490200377989218   \n",
       "85     5770286450515883198460    7062125951402879380625   \n",
       "86     7279694286431712038773    8918620590291610345692   \n",
       "87     9166090153477392630696   11241170954234462115338   \n",
       "88    11519324836522283552127   14141446120049026513095   \n",
       "89    14449673353755444902016   17756638573307399970930   \n",
       "90    18092260337835667031747   22255025941669814291789   \n",
       "91    22612398865312835945661   27842687506244864383096   \n",
       "92    28212022244946197365013   34771603814764673033251   \n",
       "93    35137421841992200284574   43349410685715738092838   \n",
       "94    43688543551438043098422   53951129547247609546474   \n",
       "95    54230142045709761859817   67033255708747064116389   \n",
       "96    67205146583816033597718   83150656334510766860102   \n",
       "97    83150656334510766860102  102976812343877587172799   \n",
       "98   102717058409866322970452  127328035243255436873229   \n",
       "99   126690849944416153037449  157192403373937340065436   \n",
       "100  156021848694303105845536  193764294973792840469175   \n",
       "\n",
       "                          98                        99   \\\n",
       "1                           1                         1   \n",
       "2                          50                        50   \n",
       "3                         898                       916   \n",
       "4                        9235                      9485   \n",
       "5                       65959                     68291   \n",
       "6                      364995                    380332   \n",
       "7                     1669629                   1750359   \n",
       "8                     6590210                   6946409   \n",
       "9                    23114319                  24487316   \n",
       "10                   73588810                  78325400   \n",
       "11                  216052066                 230970710   \n",
       "12                  592106520                 635606599   \n",
       "13                 1529212701                1647973637   \n",
       "14                 3750346946                4056570533   \n",
       "15                 8788148453                9539242715   \n",
       "16                19777130053               21539666328   \n",
       "17                42925900563               46902126205   \n",
       "18                90183805174               98842228261   \n",
       "19               183959492143              202220888315   \n",
       "20               365296539472              402709881799   \n",
       "21               707767673430              782415612693   \n",
       "22              1340675335722             1486041108043   \n",
       "23              2487165014643             2763984956628   \n",
       "24              4525904741533             5042273217353   \n",
       "25              8089515858925             9034452164161   \n",
       "26             14219630881129            15918291315366   \n",
       "27             24608207261659            27611460137231   \n",
       "28             41968850438506            47196726108342   \n",
       "29             70602128565286            79570795672857   \n",
       "30            117247663790051           132424491022471   \n",
       "..                        ...                       ...   \n",
       "71      257272919664745977803     309214212810216882901   \n",
       "72      335931770697495644028     404217267992489604313   \n",
       "73      437471634624388944439     526995974753799159133   \n",
       "74      568221726911098427733     685273863713656414125   \n",
       "75      736175205435082532308     888816028686005613511   \n",
       "76      951404700165014299248    1149941835011688574602   \n",
       "77     1226576356921400691567    1484159892189237772176   \n",
       "78     1577584555845720504762    1910952960709582784235   \n",
       "79     2024334192313292206910    2454746402619040315870   \n",
       "80     2591703090982426842590    3146100933381274347846   \n",
       "81     3310723945128516690604    4023179022478493872778   \n",
       "82     4220033349950460841865    5133544596983895280775   \n",
       "83     5367645284748487528525    6536368051974237347663   \n",
       "84     6813118095793868754831    8305123347426919996570   \n",
       "85     8630197992440904463294   10530881626019872615142   \n",
       "86    10910038709956984032601   13326326851436911697969   \n",
       "87    13765116801041977223356   16830644068586804597558   \n",
       "88    17333985567314390480713   21215460758848880865748   \n",
       "89    21787038601763603644884   26692057266971247975858   \n",
       "90    27333487068925226686742   33520104418774811690603   \n",
       "91    34229794117204917979805   42018236411383147341983   \n",
       "92    42789856269351189071663   52576826217674483587913   \n",
       "93    53397276519895299681866   65673400715356106371121   \n",
       "94    66520138639050296645877   81891215404061498831148   \n",
       "95    82728768532796604957656  101941606100399065396598   \n",
       "96   102717058409866322970452  126690849944416153037449   \n",
       "97   127328035243255436873229  157192403373937340065436   \n",
       "98   157584479243809410583714  194725543861235161461969   \n",
       "99   194725543861235161461969  240841629152665669312110   \n",
       "100  240250499761577776466542  297419407128909181291513   \n",
       "\n",
       "                          100  \n",
       "1                           1  \n",
       "2                          51  \n",
       "3                         933  \n",
       "4                        9753  \n",
       "5                       70677  \n",
       "6                      396244  \n",
       "7                     1834207  \n",
       "8                     7318923  \n",
       "9                    25929222  \n",
       "10                   83325131  \n",
       "11                  246784804  \n",
       "12                  681915067  \n",
       "13                 1774893033  \n",
       "14                 4385066429  \n",
       "15                10347814885  \n",
       "16                23443548839  \n",
       "17                51211136905  \n",
       "18               108254585389  \n",
       "19               222131761469  \n",
       "20               443620351678  \n",
       "21               864268083637  \n",
       "22              1645865507487  \n",
       "23              3069130562462  \n",
       "24              5612916075573  \n",
       "25             10081266305556  \n",
       "26             17804572300436  \n",
       "27             30954151476020  \n",
       "28             53028600157302  \n",
       "29             89597646297505  \n",
       "30            149428381490738  \n",
       "..                        ...  \n",
       "71      371142041370557573651  \n",
       "72      485725086035849614093  \n",
       "73      633976520380108981780  \n",
       "74      825307130890260352485  \n",
       "75     1071628654104984651125  \n",
       "76     1387985356857671262974  \n",
       "77     1793337098558786453476  \n",
       "78     2311528363295492026655  \n",
       "79     2972485198440800007878  \n",
       "80     3813690971111761213960  \n",
       "81     4882002649836878450104  \n",
       "82     6235882287161873662121  \n",
       "83     7948133933474026773782  \n",
       "84    10109254842978271333700  \n",
       "85    12831532115865869261911  \n",
       "86    16254042536775417511870  \n",
       "87    20548745117449190991672  \n",
       "88    25927893671751287642590  \n",
       "89    32653041745836311514237  \n",
       "90    41045965689724323609166  \n",
       "91    51501895102005980475789  \n",
       "92    64505515080297080498839  \n",
       "93    80650293735239459932684  \n",
       "94   100661793694507454764302  \n",
       "95   125425750652159548322803  \n",
       "96   156021848694303105845536  \n",
       "97   193764294973792840469175  \n",
       "98   240250499761577776466542  \n",
       "99   297419407128909181291513  \n",
       "100  367621302506997599598267  \n",
       "\n",
       "[100 rows x 100 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NN = 101\n",
    "data = timed(lambda N:[[p(x, y) for y in range(1,N)] for x in range(1,N)])(NN)\n",
    "diag = [data[i][i] for i in range(NN-1)]\n",
    "pd.DataFrame(data, columns=range(1,NN), index=range(1,NN))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " 2,\n",
       " 4,\n",
       " 9,\n",
       " 20,\n",
       " 45,\n",
       " 100,\n",
       " 220,\n",
       " 478,\n",
       " 1025,\n",
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       " 240841629152665669312110,\n",
       " 367621302506997599598267]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The above calculation with `NN = 101` took about 20 seconds in a virtual machine on a Mid 2013 Macbook Air using a Python 3 kernel. A table with `NN = 301` can be generated in about the same time on the same hardware using [pypy](http://pypy.org/). See also the accompanying `multip` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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