Av(13524, 13542, 15324, 15342, 15432, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3135, 17574, 101328, 597530, 3588409, 21873654, 134993745, 841834258, ...
Implicit Equation for the Generating Function
\(\displaystyle 2 x \left(x^{2}+x -1\right) \left(4 x^{3}-9 x^{2}+5 x -1\right) F \left(x
\right)^{6}+\left(12 x^{6}-64 x^{5}+177 x^{4}-200 x^{3}+99 x^{2}-22 x +1\right) F \left(x
\right)^{5}+\left(4 x^{6}+36 x^{5}-219 x^{4}+361 x^{3}-231 x^{2}+71 x -5\right) F \left(x
\right)^{4}+\left(32 x^{5}+12 x^{4}-204 x^{3}+202 x^{2}-104 x +10\right) F \left(x
\right)^{3}+\left(2 x^{5}+24 x^{4}+20 x^{3}-30 x^{2}+76 x -10\right) F \left(x
\right)^{2}+\left(23 x^{4}-28 x^{3}-45 x^{2}-26 x +5\right) F \! \left(x \right)-x^{4}+25 x^{3}+17 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3135\)
\(\displaystyle a(8) = 17574\)
\(\displaystyle a(9) = 101328\)
\(\displaystyle a(10) = 597530\)
\(\displaystyle a(11) = 3588409\)
\(\displaystyle a(12) = 21873654\)
\(\displaystyle a(13) = 134993745\)
\(\displaystyle a(14) = 841834258\)
\(\displaystyle a(15) = 5296570762\)
\(\displaystyle a(16) = 33580730028\)
\(\displaystyle a(17) = 214331723413\)
\(\displaystyle a(18) = 1376050874888\)
\(\displaystyle a(19) = 8880640683475\)
\(\displaystyle a(20) = 57580112794566\)
\(\displaystyle a(21) = 374897792401784\)
\(\displaystyle a(22) = 2450130035119502\)
\(\displaystyle a(23) = 16067541744133321\)
\(\displaystyle a(24) = 105696865568773034\)
\(\displaystyle a(25) = 697288027510466065\)
\(\displaystyle a(26) = 4612101131893556690\)
\(\displaystyle a(27) = 30579675419155974198\)
\(\displaystyle a(28) = 203205444324643839392\)
\(\displaystyle a(29) = 1353118432761393355993\)
\(\displaystyle a(30) = 9027578364497044912476\)
\(\displaystyle a(31) = 60337191656874456003535\)
\(\displaystyle a(32) = 403948715293159877598662\)
\(\displaystyle a(33) = 2708622526316017679981744\)
\(\displaystyle a(34) = 18189045954398904002834962\)
\(\displaystyle a(35) = 122312924665155667222690177\)
\(\displaystyle a(36) = 823570749826038111089314622\)
\(\displaystyle a(37) = 5552181949649519998744615761\)
\(\displaystyle a(38) = 37474114860594880255222802050\)
\(\displaystyle a(39) = 253207778261915658334755501642\)
\(\displaystyle a(40) = 1712678377742135454779531536340\)
\(\displaystyle a(41) = 11595907334988437652055208460525\)
\(\displaystyle a(42) = 78585522246747337240357595699200\)
\(\displaystyle a(43) = 533051920017652752521187228027123\)
\(\displaystyle a(44) = 3618822723523716977088180065650566\)
\(\displaystyle a(45) = 24587754383214120330689437953706664\)
\(\displaystyle a(46) = 167189218514148065694686319034704166\)
\(\displaystyle a(47) = 1137681371824916026778518740332168001\)
\(\displaystyle a(48) = 7747152360857139335547619557044869074\)
\(\displaystyle a(49) = 52790963685777376184105471072603157153\)
\(\displaystyle a(50) = 359965520700963497899376931129473256354\)
\(\displaystyle a(51) = 2456035038154862423972105186139199101542\)
\(\displaystyle a(52) = 16767555597716716094616272558104191152616\)
\(\displaystyle a{\left(n + 53 \right)} = \frac{1662605625 n \left(n + 1\right) \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{68 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{25 \left(n + 1\right) \left(n + 2\right) \left(2 n + 3\right) \left(378460651211 n^{2} + 12076261762408 n + 19131742321370\right) a{\left(n + 1 \right)}}{51408 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(n + 2\right) \left(112739206974508361 n^{4} + 1483043607846006228 n^{3} + 7452297527770000814 n^{2} + 16658010918635884147 n + 13843725949156371030\right) a{\left(n + 2 \right)}}{205632 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(17206 n^{3} + 2623523 n^{2} + 133283645 n + 2256132040\right) a{\left(n + 52 \right)}}{135 \left(n + 52\right) \left(n + 54\right) \left(2 n + 101\right)} - \frac{\left(105319378 n^{5} + 26284162820 n^{4} + 2623584972455 n^{3} + 130924904520400 n^{2} + 3266453863479792 n + 32594841428548800\right) a{\left(n + 51 \right)}}{15300 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(8533767488 n^{5} + 2115649306180 n^{4} + 209763503766065 n^{3} + 10397052813237260 n^{2} + 257623443593159702 n + 2552972246631728085\right) a{\left(n + 50 \right)}}{91800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(65080200952 n^{5} + 17141035544180 n^{4} + 1792801266902460 n^{3} + 93177289163872900 n^{2} + 2408461612564283783 n + 24786107104462412070\right) a{\left(n + 49 \right)}}{183600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(70341969077206 n^{5} + 15716977476876140 n^{4} + 1399483052083173455 n^{3} + 62045544744315326350 n^{2} + 1368808815467856857964 n + 12012677703936216467055\right) a{\left(n + 48 \right)}}{7711200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(2583528817963702 n^{5} + 589560725672198840 n^{4} + 53774245025486270420 n^{3} + 2450441667289870736995 n^{2} + 55785458223275421410463 n + 507545489750764788173160\right) a{\left(n + 47 \right)}}{15422400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(4053002766232324 n^{5} + 975278508515649190 n^{4} + 93425612985814725260 n^{3} + 4455831582214396396165 n^{2} + 105854075025229690974031 n + 1002424931033180778863310\right) a{\left(n + 46 \right)}}{5140800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(965142544188287332 n^{5} + 196838643179098982240 n^{4} + 15946072104167889434555 n^{3} + 640582751744219514549085 n^{2} + 12739254367629593750087418 n + 100106726358936588627471120\right) a{\left(n + 45 \right)}}{61689600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(8211457790992487337 n^{5} + 1710242204213484251890 n^{4} + 142237110647044974017370 n^{3} + 5903875314121663890080810 n^{2} + 122282127668410032032542278 n + 1010887845588427688027836185\right) a{\left(n + 44 \right)}}{20563200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(15000093258922922696 n^{5} + 401750899735090976005 n^{4} + 4063163244188827486015 n^{3} + 19501137409560852918680 n^{2} + 44757876200681199287904 n + 39569215034082985488720\right) a{\left(n + 3 \right)}}{2467584 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(1275741933713329962566 n^{5} + 263360646287514323353780 n^{4} + 21727624234611991391866525 n^{3} + 895437354913397020521288890 n^{2} + 18432984071664118269360041514 n + 151620072825480116077864420200\right) a{\left(n + 43 \right)}}{246758400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(4807679872676003110937 n^{5} + 94298178310190964683098 n^{4} + 697269526645523714376259 n^{3} + 2339175213895981776952838 n^{2} + 3207648749019936418209924 n + 810574348659021464127984\right) a{\left(n + 4 \right)}}{9870336 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(23646687607136689778584 n^{5} + 4812194728265684969890340 n^{4} + 391503291590295778286092445 n^{3} + 15916438047479321829009545920 n^{2} + 323342185659727816615321973106 n + 2625798718250650828979758709145\right) a{\left(n + 42 \right)}}{493516800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(49882176998890547891511 n^{5} + 1405498875596615517950766 n^{4} + 15808332851403615715525672 n^{3} + 88807570034928973911959783 n^{2} + 249436112280691415208279594 n + 280496361044245506518873688\right) a{\left(n + 5 \right)}}{3290112 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(342861693637449187332112 n^{5} + 68633001678646780860841340 n^{4} + 5493414702000295056456209510 n^{3} + 219760862530574037450265855690 n^{2} + 4393892837119380576368540503623 n + 35125425664918581787609743797670\right) a{\left(n + 41 \right)}}{987033600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(2763470502925761847811441 n^{5} + 91778473925919070036572724 n^{4} + 1213433995388478866044480955 n^{3} + 7993431923906063627494203680 n^{2} + 26264266933700834728999332224 n + 34467678105273476725080200856\right) a{\left(n + 6 \right)}}{32901120 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(4026330228951280834656358 n^{5} + 791921235481060416172796900 n^{4} + 62285477288028293431153433945 n^{3} + 2448663896983176971973283435690 n^{2} + 48117614509252443751033601805612 n + 378090745207308941022865509927165\right) a{\left(n + 40 \right)}}{1974067200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(38756237620148130673632182 n^{5} + 7486662977995995374150225800 n^{4} + 578322464343972093250532618410 n^{3} + 22330271779514567807115525912665 n^{2} + 430979784539011255398487308180213 n + 3326175847696536017245531704076560\right) a{\left(n + 39 \right)}}{3948134400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(50685681161171525547339604 n^{5} + 9621332494173151700829465170 n^{4} + 730255594979952310513425881125 n^{3} + 27702215565893539955360938231755 n^{2} + 525235361264659461785046480255671 n + 3981831052898029872366412427793075\right) a{\left(n + 38 \right)}}{1316044800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(314019464321076849327678648 n^{5} + 58739767740414506860826438310 n^{4} + 4391730409094643099491978907795 n^{3} + 164054546449074422373699570752980 n^{2} + 3061965555842262295289364848780177 n + 22843941124786256248821839917070910\right) a{\left(n + 37 \right)}}{2632089600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(826952162575495934226198253 n^{5} + 154231477816211532670443068330 n^{4} + 11478992227601481327188821725506 n^{3} + 426253306153900591464043128190516 n^{2} + 7898398725852488032890589007417541 n + 58434902493815530634737715913453606\right) a{\left(n + 36 \right)}}{3158507520 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(1957545448089597468742176137 n^{5} + 73957262673706632548774747125 n^{4} + 1118297648155825132375920880870 n^{3} + 8459740414392102038084034869060 n^{2} + 32019717886831047585864461956428 n + 48516809842892696966508331791000\right) a{\left(n + 7 \right)}}{987033600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(3219624915780848615843035259 n^{5} + 680756799720193689995000825320 n^{4} + 55663370450496942486138091648225 n^{3} + 2221554931997307812340245591771060 n^{2} + 43538483903488523830497652431681336 n + 336549126312221720245279739067322800\right) a{\left(n + 35 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(7824527321515378305262244073 n^{5} + 365352837214022489362278200465 n^{4} + 6775861942159842198128838154405 n^{3} + 62398864682661978701604685429695 n^{2} + 285398610604497050098743526360002 n + 518810892395295212078014785638400\right) a{\left(n + 8 \right)}}{1316044800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(22763156941237116438226290283 n^{5} + 3483090295229830516078894620860 n^{4} + 211054150167928360192323452748755 n^{3} + 6315581004571052642028613912948960 n^{2} + 93025045799053564729588407723618162 n + 536987250031586704888343185820789820\right) a{\left(n + 34 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(95069627098756523420018041913 n^{5} + 15028262682456359412446912167980 n^{4} + 948982962114467691718847135109375 n^{3} + 29920049361066961987850069301897140 n^{2} + 470952852587825177372974285761906032 n + 2960366534628513389587805151552367560\right) a{\left(n + 33 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(102425693325597780624680359963 n^{5} + 4703844354340169130506732551065 n^{4} + 83834726036546076852376487367640 n^{3} + 714761679576120690351504620189925 n^{2} + 2840393549827511990290150649925817 n + 3966375982894147985155701623694270\right) a{\left(n + 10 \right)}}{2632089600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(237820270274111025666472010999 n^{5} + 11129391281947578114755580767580 n^{4} + 208434142068617013830579862243685 n^{3} + 1950687222132231649695211236095660 n^{2} + 9114329684202931451725975079926636 n + 16994896324065066294192536860040880\right) a{\left(n + 9 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(500405485679491336546665845494 n^{5} + 78243529983018803709436470016850 n^{4} + 4891249433862950929237583505019265 n^{3} + 152807393130543419083404174356936680 n^{2} + 2385721790224967384742156113100295041 n + 14891324392028808573647728439757980430\right) a{\left(n + 32 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(833069254850913202633974996209 n^{5} + 128106054373105516979128948405420 n^{4} + 7876018464055751990843402813394755 n^{3} + 241997769121296038273861929029146780 n^{2} + 3716138339112720877019356426988040716 n + 22816335615631881645560095400723361840\right) a{\left(n + 31 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(2172160575830286356584192195949 n^{5} + 330447488989468486600561868620270 n^{4} + 20080705783197592169393726398117705 n^{3} + 609365271784223987422495561838754410 n^{2} + 9235061163898083047539393603273515186 n + 55923190473552375607667552079201648960\right) a{\left(n + 30 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(3532574322385461988132065017147 n^{5} + 566278110267602074790520763535500 n^{4} + 35843476471339289158094945647079185 n^{3} + 1122648050501914050565026120990098420 n^{2} + 17431607234024834200817612505217674108 n + 107495568718495025563440731181215967120\right) a{\left(n + 29 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(3836072071999315254024114472967 n^{5} + 457196315607245591023727466336490 n^{4} + 21269706987337561073148924548708545 n^{3} + 478015707654084365150596854680833910 n^{2} + 5098736883624750613095364684900952688 n + 19914708572856012126864911915042744400\right) a{\left(n + 28 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(4038826587624460989708072980288 n^{5} + 266863714185047321242962388939264 n^{4} + 6984674835792803143693896274165501 n^{3} + 90680313035406082818592973852352134 n^{2} + 584701959094931901124748015349213267 n + 1499274677355250035749721635570650002\right) a{\left(n + 12 \right)}}{3158507520 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(9033217434338823821311019306921 n^{5} + 623434959791146010888249917404520 n^{4} + 18933426300173170074907718131925335 n^{3} + 306340519699114848939697580573557280 n^{2} + 2556205398155187621147343933885143384 n + 8603818028025905872612992759858571320\right) a{\left(n + 13 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(11691428807694720696769257076867 n^{5} + 750734864826394053232346464638080 n^{4} + 19059018673482938698799189966864045 n^{3} + 239857686146580551411352386135292220 n^{2} + 1499599584589901107541017851404702508 n + 3731787378428646103474623569458791120\right) a{\left(n + 11 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(34110548523947935484772420106577 n^{5} + 1758726249131272456987438061582680 n^{4} - 3292763007725145775598720955612785 n^{3} - 1688762935308099644177072724865163620 n^{2} - 34567753352654691379013230647454862292 n - 214393208170953098496438448977057548400\right) a{\left(n + 23 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(39709710630040434187795978104843 n^{5} + 2895924643531105059338841966272820 n^{4} + 85253032643100591824147511763418595 n^{3} + 1265314111860252104200932425173843720 n^{2} + 9458648949670931183735840319525259682 n + 28460809828130578087257418342991553780\right) a{\left(n + 14 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(40639965623861055835335547778287 n^{5} + 5151996357675536904556897742312360 n^{4} + 260500461721906465701568082675846405 n^{3} + 6565090060129464605584037758854249640 n^{2} + 82438276244521624929729845663869475628 n + 412470733540674300346655518778433789840\right) a{\left(n + 27 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(49966743046995594356963702400941 n^{5} + 6243706317975678442965031897214740 n^{4} + 311533882226151626838218538281590375 n^{3} + 7758276897948853024347819107859022460 n^{2} + 96428269131372305206749608165231069004 n + 478509648443058572726296700110775595480\right) a{\left(n + 26 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(53838317987652013915168609482653 n^{5} + 6585232270498413744779304271850440 n^{4} + 321387577299753254076625768739164435 n^{3} + 7823842700472624428858618784107185840 n^{2} + 95012562190934394519684343698017798992 n + 460502761886831306388495904559795093880\right) a{\left(n + 25 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(76419953745150788191433819263478 n^{5} + 9319315011068946500514235596291340 n^{4} + 451142169751302961085098086295801075 n^{3} + 10846813090564036976558281362695671910 n^{2} + 129616092099118272016176832370003089947 n + 616192748353943132763639630048209721810\right) a{\left(n + 24 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(78395890156875189688859676297811 n^{5} + 7737552632387607966005142404915370 n^{4} + 305975942656362676156196929557912995 n^{3} + 6064750091175580551341955394714523190 n^{2} + 60311881197615555333143885569286750994 n + 241007146470814929365268409293705965400\right) a{\left(n + 22 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(161049500566858595277994408882601 n^{5} + 12591077093984417800198282220502800 n^{4} + 396614104281775378145962746145795035 n^{3} + 6286023376154208465329837526321256920 n^{2} + 50081329666294448690052082881555057404 n + 160310869110901620042662733153561213840\right) a{\left(n + 15 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(182686103656359578563678281247588 n^{5} + 16196169999402776440919608098166820 n^{4} + 575093189979715735915556617449895715 n^{3} + 10208571888032845096778691356886745390 n^{2} + 90501816933137160591654511095847337577 n + 320343672694167705245104038335174404110\right) a{\left(n + 16 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(273320227982015882063911253171717 n^{5} + 16984601710375248379893323527414060 n^{4} + 360045605620770220888482215276215075 n^{3} + 2343970287795376049613885230933196260 n^{2} - 12831479622364144013479052379947861472 n - 157131302081202756199898795946287047320\right) a{\left(n + 17 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(329605367644189515344038886911631 n^{5} + 28829496600493772951793768938898292 n^{4} + 1007171694633909283323315534737843365 n^{3} + 17566293960239161320316172165026462532 n^{2} + 152948740646351749828690982378820475812 n + 531830171391782560812111243693652655184\right) a{\left(n + 19 \right)}}{6317015040 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(559226880664445587906902083444869 n^{5} + 45326285284781963378727328756998530 n^{4} + 1457536804119486568052473107016949555 n^{3} + 23203798255784590236618977498584848670 n^{2} + 182455049678563490051919468536334597096 n + 565040132037976421615395177424538721800\right) a{\left(n + 18 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(794435760584668972915548040698727 n^{5} + 73440543565403802689837705052992060 n^{4} + 2717333228318495920143487286473823135 n^{3} + 50314414564014911225137181721288626920 n^{2} + 466340763098438186494983377047740277698 n + 1731416602336566148854357327435086341740\right) a{\left(n + 20 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(1080025760005264284804330931162211 n^{5} + 104296367757889116676072938608654380 n^{4} + 4035329887877406666810170830944477625 n^{3} + 78226445191132436769359533267088789180 n^{2} + 760139071607298069858638050421491367604 n + 2963504889812715931602433647749325984480\right) a{\left(n + 21 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)}, \quad n \geq 53\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3135\)
\(\displaystyle a(8) = 17574\)
\(\displaystyle a(9) = 101328\)
\(\displaystyle a(10) = 597530\)
\(\displaystyle a(11) = 3588409\)
\(\displaystyle a(12) = 21873654\)
\(\displaystyle a(13) = 134993745\)
\(\displaystyle a(14) = 841834258\)
\(\displaystyle a(15) = 5296570762\)
\(\displaystyle a(16) = 33580730028\)
\(\displaystyle a(17) = 214331723413\)
\(\displaystyle a(18) = 1376050874888\)
\(\displaystyle a(19) = 8880640683475\)
\(\displaystyle a(20) = 57580112794566\)
\(\displaystyle a(21) = 374897792401784\)
\(\displaystyle a(22) = 2450130035119502\)
\(\displaystyle a(23) = 16067541744133321\)
\(\displaystyle a(24) = 105696865568773034\)
\(\displaystyle a(25) = 697288027510466065\)
\(\displaystyle a(26) = 4612101131893556690\)
\(\displaystyle a(27) = 30579675419155974198\)
\(\displaystyle a(28) = 203205444324643839392\)
\(\displaystyle a(29) = 1353118432761393355993\)
\(\displaystyle a(30) = 9027578364497044912476\)
\(\displaystyle a(31) = 60337191656874456003535\)
\(\displaystyle a(32) = 403948715293159877598662\)
\(\displaystyle a(33) = 2708622526316017679981744\)
\(\displaystyle a(34) = 18189045954398904002834962\)
\(\displaystyle a(35) = 122312924665155667222690177\)
\(\displaystyle a(36) = 823570749826038111089314622\)
\(\displaystyle a(37) = 5552181949649519998744615761\)
\(\displaystyle a(38) = 37474114860594880255222802050\)
\(\displaystyle a(39) = 253207778261915658334755501642\)
\(\displaystyle a(40) = 1712678377742135454779531536340\)
\(\displaystyle a(41) = 11595907334988437652055208460525\)
\(\displaystyle a(42) = 78585522246747337240357595699200\)
\(\displaystyle a(43) = 533051920017652752521187228027123\)
\(\displaystyle a(44) = 3618822723523716977088180065650566\)
\(\displaystyle a(45) = 24587754383214120330689437953706664\)
\(\displaystyle a(46) = 167189218514148065694686319034704166\)
\(\displaystyle a(47) = 1137681371824916026778518740332168001\)
\(\displaystyle a(48) = 7747152360857139335547619557044869074\)
\(\displaystyle a(49) = 52790963685777376184105471072603157153\)
\(\displaystyle a(50) = 359965520700963497899376931129473256354\)
\(\displaystyle a(51) = 2456035038154862423972105186139199101542\)
\(\displaystyle a(52) = 16767555597716716094616272558104191152616\)
\(\displaystyle a{\left(n + 53 \right)} = \frac{1662605625 n \left(n + 1\right) \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{68 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{25 \left(n + 1\right) \left(n + 2\right) \left(2 n + 3\right) \left(378460651211 n^{2} + 12076261762408 n + 19131742321370\right) a{\left(n + 1 \right)}}{51408 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(n + 2\right) \left(112739206974508361 n^{4} + 1483043607846006228 n^{3} + 7452297527770000814 n^{2} + 16658010918635884147 n + 13843725949156371030\right) a{\left(n + 2 \right)}}{205632 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(17206 n^{3} + 2623523 n^{2} + 133283645 n + 2256132040\right) a{\left(n + 52 \right)}}{135 \left(n + 52\right) \left(n + 54\right) \left(2 n + 101\right)} - \frac{\left(105319378 n^{5} + 26284162820 n^{4} + 2623584972455 n^{3} + 130924904520400 n^{2} + 3266453863479792 n + 32594841428548800\right) a{\left(n + 51 \right)}}{15300 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(8533767488 n^{5} + 2115649306180 n^{4} + 209763503766065 n^{3} + 10397052813237260 n^{2} + 257623443593159702 n + 2552972246631728085\right) a{\left(n + 50 \right)}}{91800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(65080200952 n^{5} + 17141035544180 n^{4} + 1792801266902460 n^{3} + 93177289163872900 n^{2} + 2408461612564283783 n + 24786107104462412070\right) a{\left(n + 49 \right)}}{183600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(70341969077206 n^{5} + 15716977476876140 n^{4} + 1399483052083173455 n^{3} + 62045544744315326350 n^{2} + 1368808815467856857964 n + 12012677703936216467055\right) a{\left(n + 48 \right)}}{7711200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(2583528817963702 n^{5} + 589560725672198840 n^{4} + 53774245025486270420 n^{3} + 2450441667289870736995 n^{2} + 55785458223275421410463 n + 507545489750764788173160\right) a{\left(n + 47 \right)}}{15422400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(4053002766232324 n^{5} + 975278508515649190 n^{4} + 93425612985814725260 n^{3} + 4455831582214396396165 n^{2} + 105854075025229690974031 n + 1002424931033180778863310\right) a{\left(n + 46 \right)}}{5140800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(965142544188287332 n^{5} + 196838643179098982240 n^{4} + 15946072104167889434555 n^{3} + 640582751744219514549085 n^{2} + 12739254367629593750087418 n + 100106726358936588627471120\right) a{\left(n + 45 \right)}}{61689600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(8211457790992487337 n^{5} + 1710242204213484251890 n^{4} + 142237110647044974017370 n^{3} + 5903875314121663890080810 n^{2} + 122282127668410032032542278 n + 1010887845588427688027836185\right) a{\left(n + 44 \right)}}{20563200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(15000093258922922696 n^{5} + 401750899735090976005 n^{4} + 4063163244188827486015 n^{3} + 19501137409560852918680 n^{2} + 44757876200681199287904 n + 39569215034082985488720\right) a{\left(n + 3 \right)}}{2467584 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(1275741933713329962566 n^{5} + 263360646287514323353780 n^{4} + 21727624234611991391866525 n^{3} + 895437354913397020521288890 n^{2} + 18432984071664118269360041514 n + 151620072825480116077864420200\right) a{\left(n + 43 \right)}}{246758400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(4807679872676003110937 n^{5} + 94298178310190964683098 n^{4} + 697269526645523714376259 n^{3} + 2339175213895981776952838 n^{2} + 3207648749019936418209924 n + 810574348659021464127984\right) a{\left(n + 4 \right)}}{9870336 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(23646687607136689778584 n^{5} + 4812194728265684969890340 n^{4} + 391503291590295778286092445 n^{3} + 15916438047479321829009545920 n^{2} + 323342185659727816615321973106 n + 2625798718250650828979758709145\right) a{\left(n + 42 \right)}}{493516800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(49882176998890547891511 n^{5} + 1405498875596615517950766 n^{4} + 15808332851403615715525672 n^{3} + 88807570034928973911959783 n^{2} + 249436112280691415208279594 n + 280496361044245506518873688\right) a{\left(n + 5 \right)}}{3290112 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(342861693637449187332112 n^{5} + 68633001678646780860841340 n^{4} + 5493414702000295056456209510 n^{3} + 219760862530574037450265855690 n^{2} + 4393892837119380576368540503623 n + 35125425664918581787609743797670\right) a{\left(n + 41 \right)}}{987033600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(2763470502925761847811441 n^{5} + 91778473925919070036572724 n^{4} + 1213433995388478866044480955 n^{3} + 7993431923906063627494203680 n^{2} + 26264266933700834728999332224 n + 34467678105273476725080200856\right) a{\left(n + 6 \right)}}{32901120 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(4026330228951280834656358 n^{5} + 791921235481060416172796900 n^{4} + 62285477288028293431153433945 n^{3} + 2448663896983176971973283435690 n^{2} + 48117614509252443751033601805612 n + 378090745207308941022865509927165\right) a{\left(n + 40 \right)}}{1974067200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(38756237620148130673632182 n^{5} + 7486662977995995374150225800 n^{4} + 578322464343972093250532618410 n^{3} + 22330271779514567807115525912665 n^{2} + 430979784539011255398487308180213 n + 3326175847696536017245531704076560\right) a{\left(n + 39 \right)}}{3948134400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(50685681161171525547339604 n^{5} + 9621332494173151700829465170 n^{4} + 730255594979952310513425881125 n^{3} + 27702215565893539955360938231755 n^{2} + 525235361264659461785046480255671 n + 3981831052898029872366412427793075\right) a{\left(n + 38 \right)}}{1316044800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(314019464321076849327678648 n^{5} + 58739767740414506860826438310 n^{4} + 4391730409094643099491978907795 n^{3} + 164054546449074422373699570752980 n^{2} + 3061965555842262295289364848780177 n + 22843941124786256248821839917070910\right) a{\left(n + 37 \right)}}{2632089600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(826952162575495934226198253 n^{5} + 154231477816211532670443068330 n^{4} + 11478992227601481327188821725506 n^{3} + 426253306153900591464043128190516 n^{2} + 7898398725852488032890589007417541 n + 58434902493815530634737715913453606\right) a{\left(n + 36 \right)}}{3158507520 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(1957545448089597468742176137 n^{5} + 73957262673706632548774747125 n^{4} + 1118297648155825132375920880870 n^{3} + 8459740414392102038084034869060 n^{2} + 32019717886831047585864461956428 n + 48516809842892696966508331791000\right) a{\left(n + 7 \right)}}{987033600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(3219624915780848615843035259 n^{5} + 680756799720193689995000825320 n^{4} + 55663370450496942486138091648225 n^{3} + 2221554931997307812340245591771060 n^{2} + 43538483903488523830497652431681336 n + 336549126312221720245279739067322800\right) a{\left(n + 35 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(7824527321515378305262244073 n^{5} + 365352837214022489362278200465 n^{4} + 6775861942159842198128838154405 n^{3} + 62398864682661978701604685429695 n^{2} + 285398610604497050098743526360002 n + 518810892395295212078014785638400\right) a{\left(n + 8 \right)}}{1316044800 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(22763156941237116438226290283 n^{5} + 3483090295229830516078894620860 n^{4} + 211054150167928360192323452748755 n^{3} + 6315581004571052642028613912948960 n^{2} + 93025045799053564729588407723618162 n + 536987250031586704888343185820789820\right) a{\left(n + 34 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(95069627098756523420018041913 n^{5} + 15028262682456359412446912167980 n^{4} + 948982962114467691718847135109375 n^{3} + 29920049361066961987850069301897140 n^{2} + 470952852587825177372974285761906032 n + 2960366534628513389587805151552367560\right) a{\left(n + 33 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(102425693325597780624680359963 n^{5} + 4703844354340169130506732551065 n^{4} + 83834726036546076852376487367640 n^{3} + 714761679576120690351504620189925 n^{2} + 2840393549827511990290150649925817 n + 3966375982894147985155701623694270\right) a{\left(n + 10 \right)}}{2632089600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(237820270274111025666472010999 n^{5} + 11129391281947578114755580767580 n^{4} + 208434142068617013830579862243685 n^{3} + 1950687222132231649695211236095660 n^{2} + 9114329684202931451725975079926636 n + 16994896324065066294192536860040880\right) a{\left(n + 9 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(500405485679491336546665845494 n^{5} + 78243529983018803709436470016850 n^{4} + 4891249433862950929237583505019265 n^{3} + 152807393130543419083404174356936680 n^{2} + 2385721790224967384742156113100295041 n + 14891324392028808573647728439757980430\right) a{\left(n + 32 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(833069254850913202633974996209 n^{5} + 128106054373105516979128948405420 n^{4} + 7876018464055751990843402813394755 n^{3} + 241997769121296038273861929029146780 n^{2} + 3716138339112720877019356426988040716 n + 22816335615631881645560095400723361840\right) a{\left(n + 31 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(2172160575830286356584192195949 n^{5} + 330447488989468486600561868620270 n^{4} + 20080705783197592169393726398117705 n^{3} + 609365271784223987422495561838754410 n^{2} + 9235061163898083047539393603273515186 n + 55923190473552375607667552079201648960\right) a{\left(n + 30 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(3532574322385461988132065017147 n^{5} + 566278110267602074790520763535500 n^{4} + 35843476471339289158094945647079185 n^{3} + 1122648050501914050565026120990098420 n^{2} + 17431607234024834200817612505217674108 n + 107495568718495025563440731181215967120\right) a{\left(n + 29 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(3836072071999315254024114472967 n^{5} + 457196315607245591023727466336490 n^{4} + 21269706987337561073148924548708545 n^{3} + 478015707654084365150596854680833910 n^{2} + 5098736883624750613095364684900952688 n + 19914708572856012126864911915042744400\right) a{\left(n + 28 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(4038826587624460989708072980288 n^{5} + 266863714185047321242962388939264 n^{4} + 6984674835792803143693896274165501 n^{3} + 90680313035406082818592973852352134 n^{2} + 584701959094931901124748015349213267 n + 1499274677355250035749721635570650002\right) a{\left(n + 12 \right)}}{3158507520 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(9033217434338823821311019306921 n^{5} + 623434959791146010888249917404520 n^{4} + 18933426300173170074907718131925335 n^{3} + 306340519699114848939697580573557280 n^{2} + 2556205398155187621147343933885143384 n + 8603818028025905872612992759858571320\right) a{\left(n + 13 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(11691428807694720696769257076867 n^{5} + 750734864826394053232346464638080 n^{4} + 19059018673482938698799189966864045 n^{3} + 239857686146580551411352386135292220 n^{2} + 1499599584589901107541017851404702508 n + 3731787378428646103474623569458791120\right) a{\left(n + 11 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(34110548523947935484772420106577 n^{5} + 1758726249131272456987438061582680 n^{4} - 3292763007725145775598720955612785 n^{3} - 1688762935308099644177072724865163620 n^{2} - 34567753352654691379013230647454862292 n - 214393208170953098496438448977057548400\right) a{\left(n + 23 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(39709710630040434187795978104843 n^{5} + 2895924643531105059338841966272820 n^{4} + 85253032643100591824147511763418595 n^{3} + 1265314111860252104200932425173843720 n^{2} + 9458648949670931183735840319525259682 n + 28460809828130578087257418342991553780\right) a{\left(n + 14 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(40639965623861055835335547778287 n^{5} + 5151996357675536904556897742312360 n^{4} + 260500461721906465701568082675846405 n^{3} + 6565090060129464605584037758854249640 n^{2} + 82438276244521624929729845663869475628 n + 412470733540674300346655518778433789840\right) a{\left(n + 27 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(49966743046995594356963702400941 n^{5} + 6243706317975678442965031897214740 n^{4} + 311533882226151626838218538281590375 n^{3} + 7758276897948853024347819107859022460 n^{2} + 96428269131372305206749608165231069004 n + 478509648443058572726296700110775595480\right) a{\left(n + 26 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(53838317987652013915168609482653 n^{5} + 6585232270498413744779304271850440 n^{4} + 321387577299753254076625768739164435 n^{3} + 7823842700472624428858618784107185840 n^{2} + 95012562190934394519684343698017798992 n + 460502761886831306388495904559795093880\right) a{\left(n + 25 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(76419953745150788191433819263478 n^{5} + 9319315011068946500514235596291340 n^{4} + 451142169751302961085098086295801075 n^{3} + 10846813090564036976558281362695671910 n^{2} + 129616092099118272016176832370003089947 n + 616192748353943132763639630048209721810\right) a{\left(n + 24 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(78395890156875189688859676297811 n^{5} + 7737552632387607966005142404915370 n^{4} + 305975942656362676156196929557912995 n^{3} + 6064750091175580551341955394714523190 n^{2} + 60311881197615555333143885569286750994 n + 241007146470814929365268409293705965400\right) a{\left(n + 22 \right)}}{5264179200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(161049500566858595277994408882601 n^{5} + 12591077093984417800198282220502800 n^{4} + 396614104281775378145962746145795035 n^{3} + 6286023376154208465329837526321256920 n^{2} + 50081329666294448690052082881555057404 n + 160310869110901620042662733153561213840\right) a{\left(n + 15 \right)}}{10528358400 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(182686103656359578563678281247588 n^{5} + 16196169999402776440919608098166820 n^{4} + 575093189979715735915556617449895715 n^{3} + 10208571888032845096778691356886745390 n^{2} + 90501816933137160591654511095847337577 n + 320343672694167705245104038335174404110\right) a{\left(n + 16 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(273320227982015882063911253171717 n^{5} + 16984601710375248379893323527414060 n^{4} + 360045605620770220888482215276215075 n^{3} + 2343970287795376049613885230933196260 n^{2} - 12831479622364144013479052379947861472 n - 157131302081202756199898795946287047320\right) a{\left(n + 17 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(329605367644189515344038886911631 n^{5} + 28829496600493772951793768938898292 n^{4} + 1007171694633909283323315534737843365 n^{3} + 17566293960239161320316172165026462532 n^{2} + 152948740646351749828690982378820475812 n + 531830171391782560812111243693652655184\right) a{\left(n + 19 \right)}}{6317015040 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(559226880664445587906902083444869 n^{5} + 45326285284781963378727328756998530 n^{4} + 1457536804119486568052473107016949555 n^{3} + 23203798255784590236618977498584848670 n^{2} + 182455049678563490051919468536334597096 n + 565040132037976421615395177424538721800\right) a{\left(n + 18 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} - \frac{\left(794435760584668972915548040698727 n^{5} + 73440543565403802689837705052992060 n^{4} + 2717333228318495920143487286473823135 n^{3} + 50314414564014911225137181721288626920 n^{2} + 466340763098438186494983377047740277698 n + 1731416602336566148854357327435086341740\right) a{\left(n + 20 \right)}}{15792537600 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)} + \frac{\left(1080025760005264284804330931162211 n^{5} + 104296367757889116676072938608654380 n^{4} + 4035329887877406666810170830944477625 n^{3} + 78226445191132436769359533267088789180 n^{2} + 760139071607298069858638050421491367604 n + 2963504889812715931602433647749325984480\right) a{\left(n + 21 \right)}}{31585075200 \left(n + 51\right) \left(n + 52\right) \left(n + 54\right) \left(2 n + 99\right) \left(2 n + 101\right)}, \quad n \geq 53\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 146 rules.
Finding the specification took 16192 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_{25}\! \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_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{25}\! \left(x \right) &= x\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{131}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{129}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{124}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{6}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= -F_{123}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{25}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{122}\! \left(x \right) F_{25}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= \frac{F_{46}\! \left(x \right)}{F_{25}\! \left(x \right) F_{88}\! \left(x \right)}\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= \frac{F_{49}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{49}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{25}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= -F_{63}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= \frac{F_{58}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{0}\! \left(x \right) F_{25}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{25}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= -F_{71}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= \frac{F_{70}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{70}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{2}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{25}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{2}\! \left(x \right) F_{51}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{47}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= \frac{F_{80}\! \left(x \right)}{F_{25}\! \left(x \right) F_{84}\! \left(x \right)}\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{83}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{25}\! \left(x \right) F_{85}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{25}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{25}\! \left(x \right) F_{84}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{113}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{25}\! \left(x \right) F_{88}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= \frac{F_{98}\! \left(x \right)}{F_{111}\! \left(x \right)}\\
F_{98}\! \left(x \right) &= -F_{101}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= \frac{F_{100}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{100}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{0}\! \left(x \right) F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{105}\! \left(x \right) &= -F_{106}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{107}\! \left(x \right) &= \frac{F_{108}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{108}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{0}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{88} \left(x \right)^{2} F_{25}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{25}\! \left(x \right) F_{79}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{52}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{79}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{113}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{122}\! \left(x \right) F_{127}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{127}\! \left(x \right) &= \frac{F_{128}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{128}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{102}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{131}\! \left(x \right) &= -F_{142}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= \frac{F_{133}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= -F_{139}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{137}\! \left(x \right) &= \frac{F_{138}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{138}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{122}\! \left(x \right) F_{25}\! \left(x \right) F_{97}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 204 rules.
Finding the specification took 34970 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_{4}\! \left(x \right) F_{55}\! \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_{55}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\
F_{20}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{203}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{190}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= y x\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , 1, y\right)\\
F_{41}\! \left(x , y , z\right) &= -\frac{-F_{42}\! \left(x , y z \right) y +F_{42}\! \left(x , z\right)}{-1+y}\\
F_{42}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{28}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{51}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{177}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{53}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{54}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{55}\! \left(x \right) &= x\\
F_{56}\! \left(x , y\right) &= -\frac{-F_{57}\! \left(x , y\right) y +F_{57}\! \left(x , 1\right)}{-1+y}\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= -\frac{-F_{60}\! \left(x , y\right) y +F_{60}\! \left(x , 1\right)}{-1+y}\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{64}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{64}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{69}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{165}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{163}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{80}\! \left(x , y\right) F_{81}\! \left(x \right)\\
F_{80}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\
F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{123}\! \left(x \right) F_{55}\! \left(x \right)}\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= -F_{123}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{55}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= \frac{F_{88}\! \left(x \right)}{F_{157}\! \left(x \right) F_{55}\! \left(x \right)}\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= -F_{95}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= \frac{F_{91}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{55}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{97}\! \left(x \right) &= -F_{104}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= \frac{F_{99}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{0}\! \left(x \right) F_{107}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{107}\! \left(x \right) &= \frac{F_{108}\! \left(x \right)}{F_{4}\! \left(x \right) F_{55}\! \left(x \right)}\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= -F_{112}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= \frac{F_{111}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{111}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{102}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{123}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{129}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{124}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{126}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{124}\! \left(x \right) F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{55}\! \left(x \right) F_{84}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x , 1\right)\\
F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{9}\! \left(x \right)\\
F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{139}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{142}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , 1, y\right)\\
F_{141}\! \left(x , y , z\right) &= -\frac{-F_{62}\! \left(x , y z \right) y +F_{62}\! \left(x , z\right)}{-1+y}\\
F_{143}\! \left(x , y\right) &= F_{142}\! \left(x , y\right)+F_{145}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{143}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{145}\! \left(x , y\right) &= F_{146}\! \left(x , 1, y\right)\\
F_{146}\! \left(x , y , z\right) &= -\frac{-F_{60}\! \left(x , y z \right) y +F_{60}\! \left(x , z\right)}{-1+y}\\
F_{147}\! \left(x \right) &= F_{131}\! \left(x \right) F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{62}\! \left(x , 1\right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{107}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{55}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{158}\! \left(x \right) F_{55}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{131}\! \left(x \right) F_{159}\! \left(x \right)\\
F_{163}\! \left(x , y\right) &= F_{164}\! \left(x , y\right)\\
F_{164}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{80}\! \left(x , y\right) F_{84}\! \left(x \right)\\
F_{165}\! \left(x \right) &= \frac{F_{166}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right) F_{172}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{60}\! \left(x , 1\right)\\
F_{172}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{173}\! \left(x , y\right) &= F_{174}\! \left(x , y\right) F_{2}\! \left(x \right)\\
F_{175}\! \left(x , y\right) &= F_{174}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{176}\! \left(x , y\right) &= F_{175}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{176}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{177}\! \left(x , y\right) &= F_{178}\! \left(x , y\right)\\
F_{178}\! \left(x , y\right) &= F_{179}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{179}\! \left(x , y\right) &= -\frac{-F_{180}\! \left(x , y\right) y +F_{180}\! \left(x , 1\right)}{-1+y}\\
F_{180}\! \left(x , y\right) &= F_{181}\! \left(x \right)+F_{183}\! \left(x , y\right)\\
F_{181}\! \left(x \right) &= \frac{F_{182}\! \left(x \right)}{F_{55}\! \left(x \right)}\\
F_{182}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{183}\! \left(x , y\right) &= F_{184}\! \left(x , y\right)\\
F_{184}\! \left(x , y\right) &= F_{185}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{186}\! \left(x , y\right) &= F_{185}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{186}\! \left(x , y\right) &= F_{142}\! \left(x , y\right)\\
F_{187}\! \left(x , y\right) &= F_{188}\! \left(x , y\right)\\
F_{188}\! \left(x , y\right) &= F_{189}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{189}\! \left(x , y\right) &= -\frac{-y F_{20}\! \left(x , y\right)+F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{190}\! \left(x , y\right) &= F_{191}\! \left(x , y\right)+F_{192}\! \left(x , y\right)\\
F_{191}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{192}\! \left(x , y\right) &= F_{193}\! \left(x , y\right)\\
F_{193}\! \left(x , y\right) &= F_{194}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{194}\! \left(x , y\right) &= F_{195}\! \left(x , y\right)+F_{198}\! \left(x , y\right)\\
F_{195}\! \left(x , y\right) &= F_{196}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{197}\! \left(x , y\right) &= F_{196}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{197}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{198}\! \left(x , y\right) &= F_{199}\! \left(x , y\right)\\
F_{199}\! \left(x , y\right) &= F_{200}\! \left(x , y\right) F_{55}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{201}\! \left(x , y\right) &= F_{200}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{201}\! \left(x , y\right) &= F_{202}\! \left(x , y\right)\\
F_{202}\! \left(x , y\right) &= -\frac{y \left(F_{19}\! \left(x , 1\right)-F_{19}\! \left(x , y\right)\right)}{-1+y}\\
F_{203}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 149 rules.
Finding the specification took 42731 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_{26}\! \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_{26}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= x\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{137}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{135}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{130}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= -F_{126}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{129}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{26}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{26}\! \left(x \right) F_{48}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= \frac{F_{49}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{49}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{26}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{102}\! \left(x \right) F_{26}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{26}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{63}\! \left(x \right) &= -F_{70}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= \frac{F_{65}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{0}\! \left(x \right) F_{26}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= \frac{F_{74}\! \left(x \right)}{F_{26}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= -F_{78}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{77}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{2}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{26}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{2}\! \left(x \right) F_{62}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{53}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{26}\! \left(x \right) F_{48}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{115}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{26}\! \left(x \right) F_{88}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= \frac{F_{93}\! \left(x \right)}{F_{113}\! \left(x \right)}\\
F_{93}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= \frac{F_{95}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{95}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{88}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{0}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{26}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{103}\! \left(x \right) F_{26}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{108}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{110}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{0}\! \left(x \right) F_{100}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{88} \left(x \right)^{2} F_{26}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{26}\! \left(x \right) F_{53}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{58}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{100}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{104}\! \left(x \right) F_{115}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{2}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{26}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{0}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{26}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{133}\! \left(x \right) &= \frac{F_{134}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{134}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{26}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{137}\! \left(x \right) &= -F_{145}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= \frac{F_{139}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= -F_{38}\! \left(x \right)+F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{143}\! \left(x \right) &= \frac{F_{144}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{144}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{26}\! \left(x \right) F_{50}\! \left(x \right) F_{92}\! \left(x \right)\\
\end{align*}\)