Av(12453, 12543, 21453, 21543, 31452, 31542, 41352)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3126, 17374, 98635, 568830, 3320717, 19577014, 116356107, 696326804, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-8 x^{6}+15 x^{5}-16 x^{4}+13 x^{3}-10 x^{2}+6 x -2\right) F \left(x
\right)^{6}+\left(x^{6}-10 x^{5}+28 x^{4}-46 x^{3}+53 x^{2}-38 x +14\right) F \left(x
\right)^{5}+\left(x^{6}-7 x^{5}-4 x^{4}+50 x^{3}-99 x^{2}+92 x -40\right) F \left(x
\right)^{4}+\left(2 x^{5}-9 x^{4}-14 x^{3}+79 x^{2}-108 x +60\right) F \left(x
\right)^{3}+\left(x^{4}-3 x^{3}-23 x^{2}+62 x -50\right) F \left(x
\right)^{2}+\left(-14 x +22\right) F \! \left(x \right)-4 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3126\)
\(\displaystyle a(8) = 17374\)
\(\displaystyle a(9) = 98635\)
\(\displaystyle a(10) = 568830\)
\(\displaystyle a(11) = 3320717\)
\(\displaystyle a(12) = 19577014\)
\(\displaystyle a(13) = 116356107\)
\(\displaystyle a(14) = 696326804\)
\(\displaystyle a(15) = 4191791684\)
\(\displaystyle a(16) = 25364160068\)
\(\displaystyle a(17) = 154174416991\)
\(\displaystyle a(18) = 940936870642\)
\(\displaystyle a(19) = 5763511006885\)
\(\displaystyle a(20) = 35419534841928\)
\(\displaystyle a(21) = 218323607428448\)
\(\displaystyle a(22) = 1349433815019146\)
\(\displaystyle a(23) = 8361822087024797\)
\(\displaystyle a(24) = 51935694963204472\)
\(\displaystyle a(25) = 323276184778460816\)
\(\displaystyle a(26) = 2016320010896597384\)
\(\displaystyle a(27) = 12599849191510286397\)
\(\displaystyle a(28) = 78875006319416159244\)
\(\displaystyle a(29) = 494577912053573018941\)
\(\displaystyle a(30) = 3106051896873561222486\)
\(\displaystyle a(31) = 19535413091103474018724\)
\(\displaystyle a(32) = 123038470854251737975590\)
\(\displaystyle a(33) = 775945355650387374831293\)
\(\displaystyle a(34) = 4899629882138913838094638\)
\(\displaystyle a(35) = 30974882684151743305434025\)
\(\displaystyle a(36) = 196040009550069440565708438\)
\(\displaystyle a(37) = 1242065917706771384470023524\)
\(\displaystyle a(38) = 7877480909129650500506543272\)
\(\displaystyle a(39) = 50009471307421788812196579877\)
\(\displaystyle a(40) = 317775293688441306270831195144\)
\(\displaystyle a(41) = 2021030991857014064823901935365\)
\(\displaystyle a(42) = 12864530839527038759301855843994\)
\(\displaystyle a(43) = 81953454384625948497666149993046\)
\(\displaystyle a(44) = 522490097468490417520723373628166\)
\(\displaystyle a(45) = 3333591569254815905256726347499039\)
\(\displaystyle a(46) = 21284188202281942577166154774252994\)
\(\displaystyle a(47) = 135987744485001626018306968632768077\)
\(\displaystyle a(48) = 869418202580080230129683641559894618\)
\(\displaystyle a(49) = 5562024656675600168693501127521755679\)
\(\displaystyle a(50) = 35604251489800753943442229476590831396\)
\(\displaystyle a(51) = 228047693374577870225610717865899721704\)
\(\displaystyle a(52) = 1461487168587776642380004658734063775592\)
\(\displaystyle a(53) = 9371323714443465605238578693567192168671\)
\(\displaystyle a(54) = 60122225624120354934598632617101661897782\)
\(\displaystyle a(55) = 385912924782611819671501842312571953551109\)
\(\displaystyle a(56) = 2478312985279263455699086413023504604794664\)
\(\displaystyle a(57) = 15923124646192743631440748875905485158224208\)
\(\displaystyle a(58) = 102352595734716824795357678211367336714746982\)
\(\displaystyle a(59) = 658205171158198934764607858603344110267490693\)
\(\displaystyle a(60) = 4234570116213556741174121835657273203891513636\)
\(\displaystyle a(61) = 27254424349475676022114068031059702612955991335\)
\(\displaystyle a(62) = 175484457079191245790300257785931781541007474240\)
\(\displaystyle a(63) = 1130339293318173674601894462678632500432757389560\)
\(\displaystyle a(64) = 7283536992468951535643200912923789184632294130180\)
\(\displaystyle a(65) = 46949850359331745159232234164233173509348011910195\)
\(\displaystyle a(66) = 302746915832506617451299713360823138639050500452734\)
\(\displaystyle a(67) = 1952874508303583153117762852120958180895302288935427\)
\(\displaystyle a(68) = 12601251888284965615089150058911284281576557246679508\)
\(\displaystyle a(69) = 81338026952661440110070086339110601912291738027278322\)
\(\displaystyle a(70) = 525182404901601007860256699933865752981189453455944426\)
\(\displaystyle a(71) = 3392028315219308369230482921097763484342326382660101813\)
\(\displaystyle a(72) = 21914817891824357423053162194085402477154494623514299756\)
\(\displaystyle a(73) = 141625623591180640562167513943610166196817488966408204901\)
\(\displaystyle a(74) = 915520327891565671803924040351103530795950259787767728776\)
\(\displaystyle a(75) = 5919882463473394489149639050943036502573533359817319432942\)
\(\displaystyle a(76) = 38288996717703221058893898609198042140228316687002122722200\)
\(\displaystyle a(77) = 247712369527797988427410920639162776170174301573456490370223\)
\(\displaystyle a(78) = 1602991844304314806373236950520688779214536608227220344789530\)
\(\displaystyle a(79) = 10375811191984067257838399273611714444603690824680074021289685\)
\(\displaystyle a(80) = 67176481899090415604161159278369966440889338252755816263163520\)
\(\displaystyle a(81) = 435025100773233410490461581126357423309379020228925923715274112\)
\(\displaystyle a(82) = 2817803818372236982122492280874545122497231677498721328016673310\)
\(\displaystyle a(83) = 18255938620818386930430296099411075662748920284886571692265815165\)
\(\displaystyle a{\left(n + 84 \right)} = \frac{7 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{15360 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{7 \left(n + 2\right) \left(n + 3\right) \left(2 n + 3\right) \left(5822 n^{2} + 38499 n + 60508\right) a{\left(n + 1 \right)}}{368640 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{7 \left(n + 3\right) \left(2201246 n^{4} + 33474387 n^{3} + 187314439 n^{2} + 456452418 n + 407469240\right) a{\left(n + 2 \right)}}{1228800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(2435321 n + 192443358\right) a{\left(n + 83 \right)}}{23040 \left(n + 84\right)} - \frac{\left(55239481 n^{2} + 8687012969 n + 341520195618\right) a{\left(n + 82 \right)}}{11520 \left(n + 83\right) \left(n + 84\right)} + \frac{\left(23704567223 n^{3} + 5560539748728 n^{2} + 434775183767719 n + 11331186917154186\right) a{\left(n + 81 \right)}}{184320 \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(71301793825 n^{4} + 22169017111148 n^{3} + 2584665376381367 n^{2} + 133924596245833112 n + 2602125871985769924\right) a{\left(n + 80 \right)}}{30720 \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(533007024 n^{5} + 12388173195 n^{4} + 114178642270 n^{3} + 521738509905 n^{2} + 1182036658466 n + 1062187142520\right) a{\left(n + 3 \right)}}{1228800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(24967485494 n^{5} + 704749911215 n^{4} + 7909967163900 n^{3} + 44137196525685 n^{2} + 122455839665906 n + 135149459137320\right) a{\left(n + 4 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(416497648029 n^{5} + 13818671293650 n^{4} + 182544253841105 n^{3} + 1200513046372380 n^{2} + 3931475445046516 n + 5129619101524320\right) a{\left(n + 5 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(5039503065427 n^{5} + 192055333608735 n^{4} + 2915404837946195 n^{3} + 22045033269448905 n^{2} + 83062712364063858 n + 124790222130654480\right) a{\left(n + 6 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(27734630608297 n^{5} + 10712897294419865 n^{4} + 1655110734112119965 n^{3} + 127847653940062553215 n^{2} + 4937465625241132495338 n + 76269522614820661936440\right) a{\left(n + 79 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(260411512308073 n^{5} + 11203951257338120 n^{4} + 191916282263903435 n^{3} + 1637147750164393480 n^{2} + 6958670181083985972 n + 11794842405945181440\right) a{\left(n + 7 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(533834630214937 n^{5} + 203716595641822235 n^{4} + 31094367520759484405 n^{3} + 2372917618860258058525 n^{2} + 90537637809721642168938 n + 1381691717016628284240480\right) a{\left(n + 78 \right)}}{1843200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1426950368679958 n^{5} + 68312710590033185 n^{4} + 1298703538042981580 n^{3} + 12271085895619905475 n^{2} + 57680980061085683682 n + 107986803887059295760\right) a{\left(n + 8 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1919867850404458 n^{5} + 723723864621486155 n^{4} + 109120590253383173060 n^{3} + 8225883537922375998265 n^{2} + 310027489142328653074422 n + 4673576793011219176950360\right) a{\left(n + 77 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1929880929789727 n^{5} + 98917213617363890 n^{4} + 1940932277250306665 n^{3} + 18228644314839762310 n^{2} + 81439430552111191728 n + 136293156342196416000\right) a{\left(n + 9 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(10013823188770811 n^{5} + 3728958447603304000 n^{4} + 555386137659017108845 n^{3} + 41355350439601392324740 n^{2} + 1539560736357347622261324 n + 22923431768302975799488080\right) a{\left(n + 76 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21402538098657377 n^{5} + 8157326097529131385 n^{4} + 1236141880602390976645 n^{3} + 93146460609693703233575 n^{2} + 3491563956142268028060618 n + 52102166070902495892796080\right) a{\left(n + 74 \right)}}{1843200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(51291452468751496 n^{5} + 3023704072667046455 n^{4} + 71923991009589579590 n^{3} + 860386645161907408285 n^{2} + 5163859397602663874574 n + 12416779940189905772280\right) a{\left(n + 10 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(131505639460373701 n^{5} + 48410312027349142535 n^{4} + 7126939595920299105245 n^{3} + 524502848777988876010705 n^{2} + 19296119563821016118443014 n + 283894602636674717290951920\right) a{\left(n + 75 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(411928587088403341 n^{5} + 26396776067887817290 n^{4} + 679053864312276998935 n^{3} + 8753715269012467969530 n^{2} + 56486148761711897285184 n + 145839820252824557078400\right) a{\left(n + 11 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(661822470025170617 n^{5} + 235443316873591179235 n^{4} + 33515073767571236393755 n^{3} + 2386289074371064016959745 n^{2} + 84985584346312829259223368 n + 1211175943693528678802317800\right) a{\left(n + 73 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4106670727051210132 n^{5} + 287067247064714392055 n^{4} + 8039012551092989729210 n^{3} + 112619348907589031383225 n^{2} + 788649786230080721471178 n + 2207318556965346106301880\right) a{\left(n + 12 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(4560259281970758059 n^{5} + 351268050188473923850 n^{4} + 10805143287108142280920 n^{3} + 165797519859651208032275 n^{2} + 1268545856689347594177096 n + 3870909336881011951346760\right) a{\left(n + 13 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(13069590591923402291 n^{5} + 4592041629564069683585 n^{4} + 645509283842313968199995 n^{3} + 45380463737927935897373055 n^{2} + 1595556570911777126012499234 n + 22445514151227341651377922640\right) a{\left(n + 72 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17156929755212350883 n^{5} + 3646036236918694679800 n^{4} + 193123956945073579937245 n^{3} + 4368975969475348089636260 n^{2} + 45606521154040967348515332 n + 181323139320651808511171760\right) a{\left(n + 15 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(56565608872427351899 n^{5} + 19583272347317740103855 n^{4} + 2712356926970531306473215 n^{3} + 187868654416644199736551705 n^{2} + 6507513732176876209850061926 n + 90183430920317664260430725880\right) a{\left(n + 71 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(101176985250621002251 n^{5} + 8994005653977675548825 n^{4} + 314739364163779359553235 n^{3} + 5433619473854249816875015 n^{2} + 46370686732450848688769754 n + 156749319151071256014172920\right) a{\left(n + 14 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(157608898634114049082 n^{5} + 9741809418060821536955 n^{4} + 202303891061786035740455 n^{3} + 1172752374710793917088085 n^{2} - 9901277899515976752337797 n - 103187448078814076954407110\right) a{\left(n + 16 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(201931836175157858363 n^{5} + 73510784391365972823430 n^{4} + 10638003789191077894385749 n^{3} + 765599897175704819101960934 n^{2} + 27419904864297142696822736844 n + 391182134655584126848400562816\right) a{\left(n + 68 \right)}}{1474560 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(239902970697023298173 n^{5} + 81584628510293883317200 n^{4} + 11098493305982664444622075 n^{3} + 754955910338113268929156280 n^{2} + 25679730476652931796604256752 n + 349438375841654778466869396360\right) a{\left(n + 70 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1385245407016433682449 n^{5} + 454926188042693540230375 n^{4} + 59701889488049252977712965 n^{3} + 3913517516106077809760554865 n^{2} + 128134969299089766871328394426 n + 1676364409185703912510278695640\right) a{\left(n + 69 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(3081089970830489841804 n^{5} + 1056263823643356258560865 n^{4} + 144664894604297783119226910 n^{3} + 9895077339561117297962486035 n^{2} + 338041460305799594258757294026 n + 4614580098160624397447678219280\right) a{\left(n + 67 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(8398118579677393037149 n^{5} + 726263629990595886141137 n^{4} + 24844268422053350831586713 n^{3} + 419083000595185298014869727 n^{2} + 3472285312780630085306326938 n + 11238521502478291002154808784\right) a{\left(n + 18 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(9020827451762367829433 n^{5} + 702370112637302030565760 n^{4} + 21245420209537941538911655 n^{3} + 308022281304936973599935420 n^{2} + 2088178890890309176798962012 n + 5004865129860436520769163920\right) a{\left(n + 17 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(59773481587242604858338 n^{5} + 21018328565112257441317895 n^{4} + 2925568081308092353161227220 n^{3} + 201865935684285911585231327825 n^{2} + 6914605810215590261416225285922 n + 94164266574467777618458866609600\right) a{\left(n + 64 \right)}}{4915200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(99810003576103420384103 n^{5} + 34865492817151803538668355 n^{4} + 4837682683781497717203223075 n^{3} + 333632188382694409588734019085 n^{2} + 11445925495282116063652629503142 n + 156375548063052229854290506874400\right) a{\left(n + 65 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(100642073386357938009221 n^{5} + 9989683819070410774233799 n^{4} + 395511352109122796105157085 n^{3} + 7804373113569344688933922505 n^{2} + 76715611608423686200516279878 n + 300358645269524763922696533624\right) a{\left(n + 20 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(111832808406699409666559 n^{5} + 38185285186929929734357825 n^{4} + 5203777702312012320418247635 n^{3} + 353862266437676346061125546695 n^{2} + 12009319542297358490591828164926 n + 162752497284793178568993598395720\right) a{\left(n + 66 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(157967724460667035473677 n^{5} + 14750785260839982875306140 n^{4} + 548142386137351220744336875 n^{3} + 10123742020929269455875425600 n^{2} + 92829840078511641130749701028 n + 337610630617117362922185735840\right) a{\left(n + 19 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(272680504738361010864535 n^{5} + 28420447311899962160051060 n^{4} + 1182290691351819458732508377 n^{3} + 24530415356304464554455975688 n^{2} + 253753199087380512053918943396 n + 1046514657067792520102630436096\right) a{\left(n + 21 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(612453387759645579977643 n^{5} + 194260873690682746605376010 n^{4} + 24657603560790833406488893725 n^{3} + 1565495165538945237946463825430 n^{2} + 49712083286475070128157409737672 n + 631607802388587506146716097637520\right) a{\left(n + 63 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1948357473852206932112297 n^{5} + 270699938965702991438668526 n^{4} + 15057676690472688259804765837 n^{3} + 419144534106540301028490553416 n^{2} + 5838256211318665044428669733844 n + 32552553428076742729997560171596\right) a{\left(n + 27 \right)}}{245760 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(3192937706975322782012803 n^{5} + 346624618976339483407867265 n^{4} + 15018060126612755536267277075 n^{3} + 324492594702186373058579009455 n^{2} + 3495002093498304033912448683042 n + 15004247349664535517956723805000\right) a{\left(n + 22 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4286011058673723831691603 n^{5} + 546634800166214640406360649 n^{4} + 27920737806853093105270986785 n^{3} + 713869843460378506116561376537 n^{2} + 9135709242743505736658587437582 n + 46811959082191810142601933304440\right) a{\left(n + 25 \right)}}{2949120 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4994964459724893406359653 n^{5} + 1506354913567479137819202835 n^{4} + 181860906445521955182320351365 n^{3} + 10986929350150929160850062276625 n^{2} + 332150554153826094311614712715642 n + 4019767939830957537242426116040280\right) a{\left(n + 62 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(6986918408276877612384421 n^{5} + 791576484354214981810092290 n^{4} + 35808401710890189028960697735 n^{3} + 808225050966088246473395199370 n^{2} + 9098618613974029188334157586744 n + 40852612445231890752893360114400\right) a{\left(n + 23 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(7146905879566839115655098 n^{5} + 2074136900814403237602218477 n^{4} + 241281296817485264612899663364 n^{3} + 14062288762680835058622599863675 n^{2} + 410580698874740175616255660869258 n + 4803975192124261966955418836811360\right) a{\left(n + 59 \right)}}{2949120 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(16405798641228900181152089 n^{5} + 1967114517747542054247854275 n^{4} + 94353100679651262919949925745 n^{3} + 2262760956963627229075235673725 n^{2} + 27128613247939071658585429541286 n + 130065435470984230669599932430960\right) a{\left(n + 24 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17780793371926937368869079 n^{5} + 5241503474107107822031580135 n^{4} + 618422139334461943367933969375 n^{3} + 36504766082848264960685164009705 n^{2} + 1078076530802199170279298458100906 n + 12743075709121854338167401715209120\right) a{\left(n + 61 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(26231933621505707220389627 n^{5} + 7619526427839495508256212345 n^{4} + 885989192662363812998817131815 n^{3} + 51551633058888027020839683140295 n^{2} + 1500961189591840710780918876646598 n + 17494343707114902827150509696011000\right) a{\left(n + 60 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(36219573939086503419629457 n^{5} + 4855624320457593899398088385 n^{4} + 260675725493642652516663736805 n^{3} + 7004616443023505095671674115535 n^{2} + 94202676383546101607571877989018 n + 507219985331758815571956255375840\right) a{\left(n + 26 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(42493383019885249797340057 n^{5} + 11703713476977797616814075589 n^{4} + 1286362496623333117423910159873 n^{3} + 70510253875438949423384495791675 n^{2} + 1926976671772524354958106991773358 n + 20998706460888843611509933942756632\right) a{\left(n + 58 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(159317531402468808673401728 n^{5} + 22604785600019632613329617115 n^{4} + 1276392210103597864870196033170 n^{3} + 35820178887365329371618893919815 n^{2} + 499031404520728970459897543377512 n + 2756909986959160693069837861788540\right) a{\left(n + 30 \right)}}{7372800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(200912782953503283441561506 n^{5} + 28557186075413443219901150665 n^{4} + 1624603934351786736738580038790 n^{3} + 46238466240322530264874231357775 n^{2} + 658375499101564578444914143078464 n + 3751798615750636595338931583689880\right) a{\left(n + 28 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(287465515996601071303710551 n^{5} + 41828443915569378096730421380 n^{4} + 2424160935143525592658082940405 n^{3} + 69897891558371254971668509015900 n^{2} + 1001859712379419035969080778668324 n + 5704403334604480368112688098383360\right) a{\left(n + 31 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(393033166405383948828151747 n^{5} + 108063223513612516809827175600 n^{4} + 11877620731936555733913099373985 n^{3} + 652339951936698990862678826092960 n^{2} + 17901524170150575240786272952014108 n + 196356021397318761438930203161598400\right) a{\left(n + 57 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(546882873910585576659132779 n^{5} + 78273921359675090633531648080 n^{4} + 4478224240293092983657515053125 n^{3} + 128007112858287088558488669202340 n^{2} + 1827943537119062989876737619009356 n + 10431328326685441210208466430681680\right) a{\left(n + 29 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(558543861216036536572922704 n^{5} + 115964899590965868148610420925 n^{4} + 9174448147390738173155292767340 n^{3} + 334685457572822008308285942636885 n^{2} + 5187984329568141231373854793072946 n + 19255859480119592078715263098735920\right) a{\left(n + 51 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(640376099908495338257313203 n^{5} + 174557531018805707115459427909 n^{4} + 19029720879696244330204337050267 n^{3} + 1037091338757675356677543513896035 n^{2} + 28254148096020335259529075123437066 n + 307827429834392136210172456250404416\right) a{\left(n + 56 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(741422518615430154765632681 n^{5} + 200223672498638035608119124245 n^{4} + 21626731951716814839746493731795 n^{3} + 1167885894910398121848502760388085 n^{2} + 31531321088520239266804493734630994 n + 340489974917454285707404929551045520\right) a{\left(n + 54 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1812324412924564851395363041 n^{5} + 290216544209065923436496554805 n^{4} + 18657330952780052386614738429125 n^{3} + 601936249363631814718369058962795 n^{2} + 9746160033393731137829178822489114 n + 63355243568871312410710478072304240\right) a{\left(n + 32 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(2343641934276616417521119842 n^{5} + 706197957873560625810779361185 n^{4} + 83324387498862640190070762082190 n^{3} + 4837174424219681968012179358282135 n^{2} + 138647409826007050746756226597524888 n + 1573635195461296285953522847187749560\right) a{\left(n + 52 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4547957349467104895720358989 n^{5} + 788069706333091490171373353440 n^{4} + 54752136748748374872725395080055 n^{3} + 1905902453919155881979864636130920 n^{2} + 33230504344713587844881641083552996 n + 232104113315434278904029124183024320\right) a{\left(n + 33 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4729640317218353870568756413 n^{5} + 651969717780367189372793926750 n^{4} + 22484235985265776656073260898255 n^{3} - 578756794023498191757188300111690 n^{2} - 48006381883542216177741741924909248 n - 740161554837875193296178712355694480\right) a{\left(n + 45 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(5182146280192458766664560819 n^{5} + 1355484420567389588859911581193 n^{4} + 139195697396599031524956606135083 n^{3} + 7044860665898613981045310693426375 n^{2} + 176246821031421723033545464166288634 n + 1747389928046451786847810979540898696\right) a{\left(n + 46 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(6146278898948456222533393589 n^{5} + 1663558455011778227511688499260 n^{4} + 180111599116201659929267547049495 n^{3} + 9750501973593673288471228576101020 n^{2} + 263929321930339169211764709511477356 n + 2857637994813525449044863507769300800\right) a{\left(n + 55 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(9289010324890075319370096157 n^{5} + 2536993027326404625203274245660 n^{4} + 276734177907647700132762800324435 n^{3} + 15071842616249488980302367730142920 n^{2} + 409901783297839139922105479299676748 n + 4453869924063951447578037217290601200\right) a{\left(n + 53 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(10543227429830118936460679143 n^{5} + 1915032501581931665116758029345 n^{4} + 139162879873142677134166928848115 n^{3} + 5056593618749425947179561638890535 n^{2} + 91859494510364708436275447735660022 n + 667367597119879762369121591166205080\right) a{\left(n + 34 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(10663326350273471642579813885 n^{5} + 2544277947328406955233970462355 n^{4} + 242655475706516631516135985315813 n^{3} + 11562899903953417383942721326111821 n^{2} + 275283729761919754835770334609873118 n + 2619424069103542266876195757675687968\right) a{\left(n + 48 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(12701318954312184147187778479 n^{5} + 2426236085948736680566245200195 n^{4} + 185232355949007677931003003889935 n^{3} + 7065257647228520600456333683173245 n^{2} + 134642821411032254233627429118365546 n + 1025623763515673513027128545268421400\right) a{\left(n + 36 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17281011677562190617508659373 n^{5} + 3387656706513636483569565524939 n^{4} + 265519256569115040839215945438445 n^{3} + 10401264513903397838809105613914237 n^{2} + 203650252410747259521764362233156630 n + 1594395323814554272278840387719706840\right) a{\left(n + 38 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(21121358515757715824754235051 n^{5} + 5016687711814993559345970536645 n^{4} + 476189320076447551422080530507805 n^{3} + 22577300720273000773602124629658705 n^{2} + 534615371101343067884734284540031134 n + 5057305652563181260965908091677443320\right) a{\left(n + 49 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21388408371981072329553463199 n^{5} + 4002774803235638458732774973380 n^{4} + 299426042864122468148800529892145 n^{3} + 11190980156174211179620961106050360 n^{2} + 208975080349373068192421265857531196 n + 1559764168948571402767683815356799520\right) a{\left(n + 35 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21459420519768051849389809579 n^{5} + 4342431149725831515600797197880 n^{4} + 350104246333949706278595017049629 n^{3} + 14048375140593359669949736276453780 n^{2} + 280319569235306458199606908217162436 n + 2222841562474277682474350939247296160\right) a{\left(n + 43 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(23987917253209488594419026783 n^{5} + 5615107612001684518244754774515 n^{4} + 523830128674394877836403604838435 n^{3} + 24330154514540000453079226919769565 n^{2} + 562222938600896310722728805538282942 n + 5166327207766346570552634745782482160\right) a{\left(n + 50 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(30192700817004614597397844991 n^{5} + 5853146974236371181401044699015 n^{4} + 453592022469045112047391783566895 n^{3} + 17565524630578132610929047537097265 n^{2} + 339934758634263728954366090274761394 n + 2630121330965195189887467689967753600\right) a{\left(n + 37 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(49188635181702715568401869853 n^{5} + 11891398146362257221932921535770 n^{4} + 1147039052490976154113780982517695 n^{3} + 55194808897342453983822096343012210 n^{2} + 1325157351744111449974483762533140952 n + 12701037547921429770296210898146860320\right) a{\left(n + 47 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(56465531621928307283605508333 n^{5} + 11172813600714037895201121531985 n^{4} + 883890091018363787912066844820735 n^{3} + 34946737883623871817897595553045405 n^{2} + 690545712155467181769920839271936142 n + 5455683444992893241980179369132035280\right) a{\left(n + 39 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(62663866870786498174174542263 n^{5} + 12345250873904958089679284012485 n^{4} + 961194872793076966259597053154095 n^{3} + 36845515641248969086501891030020395 n^{2} + 691912131032808245563183093375374402 n + 5052738531460724079103683637172988120\right) a{\left(n + 44 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(67444231685717721884271145606 n^{5} + 13463113827798639432692957801045 n^{4} + 1074219509813773111401632148142580 n^{3} + 42823423150609689926624502406563595 n^{2} + 852888681878932467358515651756433974 n + 6788837007787739161412008137528025560\right) a{\left(n + 40 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(68322043521359522106440826839 n^{5} + 13838736640639641214752069922480 n^{4} + 1118983673935624358867344163731915 n^{3} + 45137203630502050418231546062251500 n^{2} + 908012218102615165796910273171773286 n + 7284898050762793170955588198198401060\right) a{\left(n + 42 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(145118434030180389368265146353 n^{5} + 29212103511690862119857974605410 n^{4} + 2349354565910547608345922293462075 n^{3} + 94348917536222235891326484833316610 n^{2} + 1891770445484823262439046628064126952 n + 15148371054165013470583000644634972080\right) a{\left(n + 41 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)}, \quad n \geq 84\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3126\)
\(\displaystyle a(8) = 17374\)
\(\displaystyle a(9) = 98635\)
\(\displaystyle a(10) = 568830\)
\(\displaystyle a(11) = 3320717\)
\(\displaystyle a(12) = 19577014\)
\(\displaystyle a(13) = 116356107\)
\(\displaystyle a(14) = 696326804\)
\(\displaystyle a(15) = 4191791684\)
\(\displaystyle a(16) = 25364160068\)
\(\displaystyle a(17) = 154174416991\)
\(\displaystyle a(18) = 940936870642\)
\(\displaystyle a(19) = 5763511006885\)
\(\displaystyle a(20) = 35419534841928\)
\(\displaystyle a(21) = 218323607428448\)
\(\displaystyle a(22) = 1349433815019146\)
\(\displaystyle a(23) = 8361822087024797\)
\(\displaystyle a(24) = 51935694963204472\)
\(\displaystyle a(25) = 323276184778460816\)
\(\displaystyle a(26) = 2016320010896597384\)
\(\displaystyle a(27) = 12599849191510286397\)
\(\displaystyle a(28) = 78875006319416159244\)
\(\displaystyle a(29) = 494577912053573018941\)
\(\displaystyle a(30) = 3106051896873561222486\)
\(\displaystyle a(31) = 19535413091103474018724\)
\(\displaystyle a(32) = 123038470854251737975590\)
\(\displaystyle a(33) = 775945355650387374831293\)
\(\displaystyle a(34) = 4899629882138913838094638\)
\(\displaystyle a(35) = 30974882684151743305434025\)
\(\displaystyle a(36) = 196040009550069440565708438\)
\(\displaystyle a(37) = 1242065917706771384470023524\)
\(\displaystyle a(38) = 7877480909129650500506543272\)
\(\displaystyle a(39) = 50009471307421788812196579877\)
\(\displaystyle a(40) = 317775293688441306270831195144\)
\(\displaystyle a(41) = 2021030991857014064823901935365\)
\(\displaystyle a(42) = 12864530839527038759301855843994\)
\(\displaystyle a(43) = 81953454384625948497666149993046\)
\(\displaystyle a(44) = 522490097468490417520723373628166\)
\(\displaystyle a(45) = 3333591569254815905256726347499039\)
\(\displaystyle a(46) = 21284188202281942577166154774252994\)
\(\displaystyle a(47) = 135987744485001626018306968632768077\)
\(\displaystyle a(48) = 869418202580080230129683641559894618\)
\(\displaystyle a(49) = 5562024656675600168693501127521755679\)
\(\displaystyle a(50) = 35604251489800753943442229476590831396\)
\(\displaystyle a(51) = 228047693374577870225610717865899721704\)
\(\displaystyle a(52) = 1461487168587776642380004658734063775592\)
\(\displaystyle a(53) = 9371323714443465605238578693567192168671\)
\(\displaystyle a(54) = 60122225624120354934598632617101661897782\)
\(\displaystyle a(55) = 385912924782611819671501842312571953551109\)
\(\displaystyle a(56) = 2478312985279263455699086413023504604794664\)
\(\displaystyle a(57) = 15923124646192743631440748875905485158224208\)
\(\displaystyle a(58) = 102352595734716824795357678211367336714746982\)
\(\displaystyle a(59) = 658205171158198934764607858603344110267490693\)
\(\displaystyle a(60) = 4234570116213556741174121835657273203891513636\)
\(\displaystyle a(61) = 27254424349475676022114068031059702612955991335\)
\(\displaystyle a(62) = 175484457079191245790300257785931781541007474240\)
\(\displaystyle a(63) = 1130339293318173674601894462678632500432757389560\)
\(\displaystyle a(64) = 7283536992468951535643200912923789184632294130180\)
\(\displaystyle a(65) = 46949850359331745159232234164233173509348011910195\)
\(\displaystyle a(66) = 302746915832506617451299713360823138639050500452734\)
\(\displaystyle a(67) = 1952874508303583153117762852120958180895302288935427\)
\(\displaystyle a(68) = 12601251888284965615089150058911284281576557246679508\)
\(\displaystyle a(69) = 81338026952661440110070086339110601912291738027278322\)
\(\displaystyle a(70) = 525182404901601007860256699933865752981189453455944426\)
\(\displaystyle a(71) = 3392028315219308369230482921097763484342326382660101813\)
\(\displaystyle a(72) = 21914817891824357423053162194085402477154494623514299756\)
\(\displaystyle a(73) = 141625623591180640562167513943610166196817488966408204901\)
\(\displaystyle a(74) = 915520327891565671803924040351103530795950259787767728776\)
\(\displaystyle a(75) = 5919882463473394489149639050943036502573533359817319432942\)
\(\displaystyle a(76) = 38288996717703221058893898609198042140228316687002122722200\)
\(\displaystyle a(77) = 247712369527797988427410920639162776170174301573456490370223\)
\(\displaystyle a(78) = 1602991844304314806373236950520688779214536608227220344789530\)
\(\displaystyle a(79) = 10375811191984067257838399273611714444603690824680074021289685\)
\(\displaystyle a(80) = 67176481899090415604161159278369966440889338252755816263163520\)
\(\displaystyle a(81) = 435025100773233410490461581126357423309379020228925923715274112\)
\(\displaystyle a(82) = 2817803818372236982122492280874545122497231677498721328016673310\)
\(\displaystyle a(83) = 18255938620818386930430296099411075662748920284886571692265815165\)
\(\displaystyle a{\left(n + 84 \right)} = \frac{7 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{15360 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{7 \left(n + 2\right) \left(n + 3\right) \left(2 n + 3\right) \left(5822 n^{2} + 38499 n + 60508\right) a{\left(n + 1 \right)}}{368640 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{7 \left(n + 3\right) \left(2201246 n^{4} + 33474387 n^{3} + 187314439 n^{2} + 456452418 n + 407469240\right) a{\left(n + 2 \right)}}{1228800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(2435321 n + 192443358\right) a{\left(n + 83 \right)}}{23040 \left(n + 84\right)} - \frac{\left(55239481 n^{2} + 8687012969 n + 341520195618\right) a{\left(n + 82 \right)}}{11520 \left(n + 83\right) \left(n + 84\right)} + \frac{\left(23704567223 n^{3} + 5560539748728 n^{2} + 434775183767719 n + 11331186917154186\right) a{\left(n + 81 \right)}}{184320 \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(71301793825 n^{4} + 22169017111148 n^{3} + 2584665376381367 n^{2} + 133924596245833112 n + 2602125871985769924\right) a{\left(n + 80 \right)}}{30720 \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(533007024 n^{5} + 12388173195 n^{4} + 114178642270 n^{3} + 521738509905 n^{2} + 1182036658466 n + 1062187142520\right) a{\left(n + 3 \right)}}{1228800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(24967485494 n^{5} + 704749911215 n^{4} + 7909967163900 n^{3} + 44137196525685 n^{2} + 122455839665906 n + 135149459137320\right) a{\left(n + 4 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(416497648029 n^{5} + 13818671293650 n^{4} + 182544253841105 n^{3} + 1200513046372380 n^{2} + 3931475445046516 n + 5129619101524320\right) a{\left(n + 5 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(5039503065427 n^{5} + 192055333608735 n^{4} + 2915404837946195 n^{3} + 22045033269448905 n^{2} + 83062712364063858 n + 124790222130654480\right) a{\left(n + 6 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(27734630608297 n^{5} + 10712897294419865 n^{4} + 1655110734112119965 n^{3} + 127847653940062553215 n^{2} + 4937465625241132495338 n + 76269522614820661936440\right) a{\left(n + 79 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(260411512308073 n^{5} + 11203951257338120 n^{4} + 191916282263903435 n^{3} + 1637147750164393480 n^{2} + 6958670181083985972 n + 11794842405945181440\right) a{\left(n + 7 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(533834630214937 n^{5} + 203716595641822235 n^{4} + 31094367520759484405 n^{3} + 2372917618860258058525 n^{2} + 90537637809721642168938 n + 1381691717016628284240480\right) a{\left(n + 78 \right)}}{1843200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1426950368679958 n^{5} + 68312710590033185 n^{4} + 1298703538042981580 n^{3} + 12271085895619905475 n^{2} + 57680980061085683682 n + 107986803887059295760\right) a{\left(n + 8 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1919867850404458 n^{5} + 723723864621486155 n^{4} + 109120590253383173060 n^{3} + 8225883537922375998265 n^{2} + 310027489142328653074422 n + 4673576793011219176950360\right) a{\left(n + 77 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1929880929789727 n^{5} + 98917213617363890 n^{4} + 1940932277250306665 n^{3} + 18228644314839762310 n^{2} + 81439430552111191728 n + 136293156342196416000\right) a{\left(n + 9 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(10013823188770811 n^{5} + 3728958447603304000 n^{4} + 555386137659017108845 n^{3} + 41355350439601392324740 n^{2} + 1539560736357347622261324 n + 22923431768302975799488080\right) a{\left(n + 76 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21402538098657377 n^{5} + 8157326097529131385 n^{4} + 1236141880602390976645 n^{3} + 93146460609693703233575 n^{2} + 3491563956142268028060618 n + 52102166070902495892796080\right) a{\left(n + 74 \right)}}{1843200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(51291452468751496 n^{5} + 3023704072667046455 n^{4} + 71923991009589579590 n^{3} + 860386645161907408285 n^{2} + 5163859397602663874574 n + 12416779940189905772280\right) a{\left(n + 10 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(131505639460373701 n^{5} + 48410312027349142535 n^{4} + 7126939595920299105245 n^{3} + 524502848777988876010705 n^{2} + 19296119563821016118443014 n + 283894602636674717290951920\right) a{\left(n + 75 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(411928587088403341 n^{5} + 26396776067887817290 n^{4} + 679053864312276998935 n^{3} + 8753715269012467969530 n^{2} + 56486148761711897285184 n + 145839820252824557078400\right) a{\left(n + 11 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(661822470025170617 n^{5} + 235443316873591179235 n^{4} + 33515073767571236393755 n^{3} + 2386289074371064016959745 n^{2} + 84985584346312829259223368 n + 1211175943693528678802317800\right) a{\left(n + 73 \right)}}{921600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4106670727051210132 n^{5} + 287067247064714392055 n^{4} + 8039012551092989729210 n^{3} + 112619348907589031383225 n^{2} + 788649786230080721471178 n + 2207318556965346106301880\right) a{\left(n + 12 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(4560259281970758059 n^{5} + 351268050188473923850 n^{4} + 10805143287108142280920 n^{3} + 165797519859651208032275 n^{2} + 1268545856689347594177096 n + 3870909336881011951346760\right) a{\left(n + 13 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(13069590591923402291 n^{5} + 4592041629564069683585 n^{4} + 645509283842313968199995 n^{3} + 45380463737927935897373055 n^{2} + 1595556570911777126012499234 n + 22445514151227341651377922640\right) a{\left(n + 72 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17156929755212350883 n^{5} + 3646036236918694679800 n^{4} + 193123956945073579937245 n^{3} + 4368975969475348089636260 n^{2} + 45606521154040967348515332 n + 181323139320651808511171760\right) a{\left(n + 15 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(56565608872427351899 n^{5} + 19583272347317740103855 n^{4} + 2712356926970531306473215 n^{3} + 187868654416644199736551705 n^{2} + 6507513732176876209850061926 n + 90183430920317664260430725880\right) a{\left(n + 71 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(101176985250621002251 n^{5} + 8994005653977675548825 n^{4} + 314739364163779359553235 n^{3} + 5433619473854249816875015 n^{2} + 46370686732450848688769754 n + 156749319151071256014172920\right) a{\left(n + 14 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(157608898634114049082 n^{5} + 9741809418060821536955 n^{4} + 202303891061786035740455 n^{3} + 1172752374710793917088085 n^{2} - 9901277899515976752337797 n - 103187448078814076954407110\right) a{\left(n + 16 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(201931836175157858363 n^{5} + 73510784391365972823430 n^{4} + 10638003789191077894385749 n^{3} + 765599897175704819101960934 n^{2} + 27419904864297142696822736844 n + 391182134655584126848400562816\right) a{\left(n + 68 \right)}}{1474560 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(239902970697023298173 n^{5} + 81584628510293883317200 n^{4} + 11098493305982664444622075 n^{3} + 754955910338113268929156280 n^{2} + 25679730476652931796604256752 n + 349438375841654778466869396360\right) a{\left(n + 70 \right)}}{3686400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1385245407016433682449 n^{5} + 454926188042693540230375 n^{4} + 59701889488049252977712965 n^{3} + 3913517516106077809760554865 n^{2} + 128134969299089766871328394426 n + 1676364409185703912510278695640\right) a{\left(n + 69 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(3081089970830489841804 n^{5} + 1056263823643356258560865 n^{4} + 144664894604297783119226910 n^{3} + 9895077339561117297962486035 n^{2} + 338041460305799594258757294026 n + 4614580098160624397447678219280\right) a{\left(n + 67 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(8398118579677393037149 n^{5} + 726263629990595886141137 n^{4} + 24844268422053350831586713 n^{3} + 419083000595185298014869727 n^{2} + 3472285312780630085306326938 n + 11238521502478291002154808784\right) a{\left(n + 18 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(9020827451762367829433 n^{5} + 702370112637302030565760 n^{4} + 21245420209537941538911655 n^{3} + 308022281304936973599935420 n^{2} + 2088178890890309176798962012 n + 5004865129860436520769163920\right) a{\left(n + 17 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(59773481587242604858338 n^{5} + 21018328565112257441317895 n^{4} + 2925568081308092353161227220 n^{3} + 201865935684285911585231327825 n^{2} + 6914605810215590261416225285922 n + 94164266574467777618458866609600\right) a{\left(n + 64 \right)}}{4915200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(99810003576103420384103 n^{5} + 34865492817151803538668355 n^{4} + 4837682683781497717203223075 n^{3} + 333632188382694409588734019085 n^{2} + 11445925495282116063652629503142 n + 156375548063052229854290506874400\right) a{\left(n + 65 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(100642073386357938009221 n^{5} + 9989683819070410774233799 n^{4} + 395511352109122796105157085 n^{3} + 7804373113569344688933922505 n^{2} + 76715611608423686200516279878 n + 300358645269524763922696533624\right) a{\left(n + 20 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(111832808406699409666559 n^{5} + 38185285186929929734357825 n^{4} + 5203777702312012320418247635 n^{3} + 353862266437676346061125546695 n^{2} + 12009319542297358490591828164926 n + 162752497284793178568993598395720\right) a{\left(n + 66 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(157967724460667035473677 n^{5} + 14750785260839982875306140 n^{4} + 548142386137351220744336875 n^{3} + 10123742020929269455875425600 n^{2} + 92829840078511641130749701028 n + 337610630617117362922185735840\right) a{\left(n + 19 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(272680504738361010864535 n^{5} + 28420447311899962160051060 n^{4} + 1182290691351819458732508377 n^{3} + 24530415356304464554455975688 n^{2} + 253753199087380512053918943396 n + 1046514657067792520102630436096\right) a{\left(n + 21 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(612453387759645579977643 n^{5} + 194260873690682746605376010 n^{4} + 24657603560790833406488893725 n^{3} + 1565495165538945237946463825430 n^{2} + 49712083286475070128157409737672 n + 631607802388587506146716097637520\right) a{\left(n + 63 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(1948357473852206932112297 n^{5} + 270699938965702991438668526 n^{4} + 15057676690472688259804765837 n^{3} + 419144534106540301028490553416 n^{2} + 5838256211318665044428669733844 n + 32552553428076742729997560171596\right) a{\left(n + 27 \right)}}{245760 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(3192937706975322782012803 n^{5} + 346624618976339483407867265 n^{4} + 15018060126612755536267277075 n^{3} + 324492594702186373058579009455 n^{2} + 3495002093498304033912448683042 n + 15004247349664535517956723805000\right) a{\left(n + 22 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4286011058673723831691603 n^{5} + 546634800166214640406360649 n^{4} + 27920737806853093105270986785 n^{3} + 713869843460378506116561376537 n^{2} + 9135709242743505736658587437582 n + 46811959082191810142601933304440\right) a{\left(n + 25 \right)}}{2949120 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4994964459724893406359653 n^{5} + 1506354913567479137819202835 n^{4} + 181860906445521955182320351365 n^{3} + 10986929350150929160850062276625 n^{2} + 332150554153826094311614712715642 n + 4019767939830957537242426116040280\right) a{\left(n + 62 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(6986918408276877612384421 n^{5} + 791576484354214981810092290 n^{4} + 35808401710890189028960697735 n^{3} + 808225050966088246473395199370 n^{2} + 9098618613974029188334157586744 n + 40852612445231890752893360114400\right) a{\left(n + 23 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(7146905879566839115655098 n^{5} + 2074136900814403237602218477 n^{4} + 241281296817485264612899663364 n^{3} + 14062288762680835058622599863675 n^{2} + 410580698874740175616255660869258 n + 4803975192124261966955418836811360\right) a{\left(n + 59 \right)}}{2949120 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(16405798641228900181152089 n^{5} + 1967114517747542054247854275 n^{4} + 94353100679651262919949925745 n^{3} + 2262760956963627229075235673725 n^{2} + 27128613247939071658585429541286 n + 130065435470984230669599932430960\right) a{\left(n + 24 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17780793371926937368869079 n^{5} + 5241503474107107822031580135 n^{4} + 618422139334461943367933969375 n^{3} + 36504766082848264960685164009705 n^{2} + 1078076530802199170279298458100906 n + 12743075709121854338167401715209120\right) a{\left(n + 61 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(26231933621505707220389627 n^{5} + 7619526427839495508256212345 n^{4} + 885989192662363812998817131815 n^{3} + 51551633058888027020839683140295 n^{2} + 1500961189591840710780918876646598 n + 17494343707114902827150509696011000\right) a{\left(n + 60 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(36219573939086503419629457 n^{5} + 4855624320457593899398088385 n^{4} + 260675725493642652516663736805 n^{3} + 7004616443023505095671674115535 n^{2} + 94202676383546101607571877989018 n + 507219985331758815571956255375840\right) a{\left(n + 26 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(42493383019885249797340057 n^{5} + 11703713476977797616814075589 n^{4} + 1286362496623333117423910159873 n^{3} + 70510253875438949423384495791675 n^{2} + 1926976671772524354958106991773358 n + 20998706460888843611509933942756632\right) a{\left(n + 58 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(159317531402468808673401728 n^{5} + 22604785600019632613329617115 n^{4} + 1276392210103597864870196033170 n^{3} + 35820178887365329371618893919815 n^{2} + 499031404520728970459897543377512 n + 2756909986959160693069837861788540\right) a{\left(n + 30 \right)}}{7372800 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(200912782953503283441561506 n^{5} + 28557186075413443219901150665 n^{4} + 1624603934351786736738580038790 n^{3} + 46238466240322530264874231357775 n^{2} + 658375499101564578444914143078464 n + 3751798615750636595338931583689880\right) a{\left(n + 28 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(287465515996601071303710551 n^{5} + 41828443915569378096730421380 n^{4} + 2424160935143525592658082940405 n^{3} + 69897891558371254971668509015900 n^{2} + 1001859712379419035969080778668324 n + 5704403334604480368112688098383360\right) a{\left(n + 31 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(393033166405383948828151747 n^{5} + 108063223513612516809827175600 n^{4} + 11877620731936555733913099373985 n^{3} + 652339951936698990862678826092960 n^{2} + 17901524170150575240786272952014108 n + 196356021397318761438930203161598400\right) a{\left(n + 57 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(546882873910585576659132779 n^{5} + 78273921359675090633531648080 n^{4} + 4478224240293092983657515053125 n^{3} + 128007112858287088558488669202340 n^{2} + 1827943537119062989876737619009356 n + 10431328326685441210208466430681680\right) a{\left(n + 29 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(558543861216036536572922704 n^{5} + 115964899590965868148610420925 n^{4} + 9174448147390738173155292767340 n^{3} + 334685457572822008308285942636885 n^{2} + 5187984329568141231373854793072946 n + 19255859480119592078715263098735920\right) a{\left(n + 51 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(640376099908495338257313203 n^{5} + 174557531018805707115459427909 n^{4} + 19029720879696244330204337050267 n^{3} + 1037091338757675356677543513896035 n^{2} + 28254148096020335259529075123437066 n + 307827429834392136210172456250404416\right) a{\left(n + 56 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(741422518615430154765632681 n^{5} + 200223672498638035608119124245 n^{4} + 21626731951716814839746493731795 n^{3} + 1167885894910398121848502760388085 n^{2} + 31531321088520239266804493734630994 n + 340489974917454285707404929551045520\right) a{\left(n + 54 \right)}}{2457600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(1812324412924564851395363041 n^{5} + 290216544209065923436496554805 n^{4} + 18657330952780052386614738429125 n^{3} + 601936249363631814718369058962795 n^{2} + 9746160033393731137829178822489114 n + 63355243568871312410710478072304240\right) a{\left(n + 32 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(2343641934276616417521119842 n^{5} + 706197957873560625810779361185 n^{4} + 83324387498862640190070762082190 n^{3} + 4837174424219681968012179358282135 n^{2} + 138647409826007050746756226597524888 n + 1573635195461296285953522847187749560\right) a{\left(n + 52 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4547957349467104895720358989 n^{5} + 788069706333091490171373353440 n^{4} + 54752136748748374872725395080055 n^{3} + 1905902453919155881979864636130920 n^{2} + 33230504344713587844881641083552996 n + 232104113315434278904029124183024320\right) a{\left(n + 33 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(4729640317218353870568756413 n^{5} + 651969717780367189372793926750 n^{4} + 22484235985265776656073260898255 n^{3} - 578756794023498191757188300111690 n^{2} - 48006381883542216177741741924909248 n - 740161554837875193296178712355694480\right) a{\left(n + 45 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(5182146280192458766664560819 n^{5} + 1355484420567389588859911581193 n^{4} + 139195697396599031524956606135083 n^{3} + 7044860665898613981045310693426375 n^{2} + 176246821031421723033545464166288634 n + 1747389928046451786847810979540898696\right) a{\left(n + 46 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(6146278898948456222533393589 n^{5} + 1663558455011778227511688499260 n^{4} + 180111599116201659929267547049495 n^{3} + 9750501973593673288471228576101020 n^{2} + 263929321930339169211764709511477356 n + 2857637994813525449044863507769300800\right) a{\left(n + 55 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(9289010324890075319370096157 n^{5} + 2536993027326404625203274245660 n^{4} + 276734177907647700132762800324435 n^{3} + 15071842616249488980302367730142920 n^{2} + 409901783297839139922105479299676748 n + 4453869924063951447578037217290601200\right) a{\left(n + 53 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(10543227429830118936460679143 n^{5} + 1915032501581931665116758029345 n^{4} + 139162879873142677134166928848115 n^{3} + 5056593618749425947179561638890535 n^{2} + 91859494510364708436275447735660022 n + 667367597119879762369121591166205080\right) a{\left(n + 34 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(10663326350273471642579813885 n^{5} + 2544277947328406955233970462355 n^{4} + 242655475706516631516135985315813 n^{3} + 11562899903953417383942721326111821 n^{2} + 275283729761919754835770334609873118 n + 2619424069103542266876195757675687968\right) a{\left(n + 48 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(12701318954312184147187778479 n^{5} + 2426236085948736680566245200195 n^{4} + 185232355949007677931003003889935 n^{3} + 7065257647228520600456333683173245 n^{2} + 134642821411032254233627429118365546 n + 1025623763515673513027128545268421400\right) a{\left(n + 36 \right)}}{9830400 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(17281011677562190617508659373 n^{5} + 3387656706513636483569565524939 n^{4} + 265519256569115040839215945438445 n^{3} + 10401264513903397838809105613914237 n^{2} + 203650252410747259521764362233156630 n + 1594395323814554272278840387719706840\right) a{\left(n + 38 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(21121358515757715824754235051 n^{5} + 5016687711814993559345970536645 n^{4} + 476189320076447551422080530507805 n^{3} + 22577300720273000773602124629658705 n^{2} + 534615371101343067884734284540031134 n + 5057305652563181260965908091677443320\right) a{\left(n + 49 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21388408371981072329553463199 n^{5} + 4002774803235638458732774973380 n^{4} + 299426042864122468148800529892145 n^{3} + 11190980156174211179620961106050360 n^{2} + 208975080349373068192421265857531196 n + 1559764168948571402767683815356799520\right) a{\left(n + 35 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(21459420519768051849389809579 n^{5} + 4342431149725831515600797197880 n^{4} + 350104246333949706278595017049629 n^{3} + 14048375140593359669949736276453780 n^{2} + 280319569235306458199606908217162436 n + 2222841562474277682474350939247296160\right) a{\left(n + 43 \right)}}{5898240 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(23987917253209488594419026783 n^{5} + 5615107612001684518244754774515 n^{4} + 523830128674394877836403604838435 n^{3} + 24330154514540000453079226919769565 n^{2} + 562222938600896310722728805538282942 n + 5166327207766346570552634745782482160\right) a{\left(n + 50 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(30192700817004614597397844991 n^{5} + 5853146974236371181401044699015 n^{4} + 453592022469045112047391783566895 n^{3} + 17565524630578132610929047537097265 n^{2} + 339934758634263728954366090274761394 n + 2630121330965195189887467689967753600\right) a{\left(n + 37 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(49188635181702715568401869853 n^{5} + 11891398146362257221932921535770 n^{4} + 1147039052490976154113780982517695 n^{3} + 55194808897342453983822096343012210 n^{2} + 1325157351744111449974483762533140952 n + 12701037547921429770296210898146860320\right) a{\left(n + 47 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(56465531621928307283605508333 n^{5} + 11172813600714037895201121531985 n^{4} + 883890091018363787912066844820735 n^{3} + 34946737883623871817897595553045405 n^{2} + 690545712155467181769920839271936142 n + 5455683444992893241980179369132035280\right) a{\left(n + 39 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(62663866870786498174174542263 n^{5} + 12345250873904958089679284012485 n^{4} + 961194872793076966259597053154095 n^{3} + 36845515641248969086501891030020395 n^{2} + 691912131032808245563183093375374402 n + 5052738531460724079103683637172988120\right) a{\left(n + 44 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(67444231685717721884271145606 n^{5} + 13463113827798639432692957801045 n^{4} + 1074219509813773111401632148142580 n^{3} + 42823423150609689926624502406563595 n^{2} + 852888681878932467358515651756433974 n + 6788837007787739161412008137528025560\right) a{\left(n + 40 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} + \frac{\left(68322043521359522106440826839 n^{5} + 13838736640639641214752069922480 n^{4} + 1118983673935624358867344163731915 n^{3} + 45137203630502050418231546062251500 n^{2} + 908012218102615165796910273171773286 n + 7284898050762793170955588198198401060\right) a{\left(n + 42 \right)}}{14745600 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)} - \frac{\left(145118434030180389368265146353 n^{5} + 29212103511690862119857974605410 n^{4} + 2349354565910547608345922293462075 n^{3} + 94348917536222235891326484833316610 n^{2} + 1891770445484823262439046628064126952 n + 15148371054165013470583000644634972080\right) a{\left(n + 41 \right)}}{29491200 \left(n + 80\right) \left(n + 81\right) \left(n + 82\right) \left(n + 83\right) \left(n + 84\right)}, \quad n \geq 84\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 78 rules.
Finding the specification took 2049 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{22}\! \left(x \right) &= x\\
F_{23}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{29}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{29}\! \left(x \right) x +F_{29} \left(x \right)^{2}-2 F_{29}\! \left(x \right)+2\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{22}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{2}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{76}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{22}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{28}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{51}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{0}\! \left(x \right) F_{22}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{22}\! \left(x \right) F_{28}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{61}\! \left(x \right) &= -F_{62}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{62}\! \left(x \right) &= -F_{63}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{22}\! \left(x \right) F_{55}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{75}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{22}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= \frac{F_{74}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{74}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{76}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 67 rules.
Finding the specification took 888 seconds.
Copy 67 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{0}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{28}\! \left(x \right) x +F_{28} \left(x \right)^{2}+x\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{34}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= -F_{64}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{17}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{17}\! \left(x \right) F_{50}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{17}\! \left(x \right) F_{55}\! \left(x \right)}\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{56}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{17}\! \left(x \right) F_{63}\! \left(x \right)}\\
F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{63}\! \left(x \right) x +F_{63} \left(x \right)^{2}-2 F_{63}\! \left(x \right)+2\\
F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 78 rules.
Finding the specification took 1815 seconds.
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Copy 78 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{25}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= x\\
F_{25}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{29}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{29}\! \left(x \right) x +F_{29} \left(x \right)^{2}+x\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{24}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{2}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{76}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= \frac{F_{46}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{24}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{33}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{0}\! \left(x \right) F_{24}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{24}\! \left(x \right) F_{33}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{61}\! \left(x \right) &= -F_{62}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{62}\! \left(x \right) &= -F_{63}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{24}\! \left(x \right) F_{56}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{75}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{24}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= \frac{F_{74}\! \left(x \right)}{F_{24}\! \left(x \right)}\\
F_{74}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{76}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
\end{align*}\)