Av(12453, 12543, 21453, 21543, 31452, 31542, 41532)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3125, 17354, 98386, 566333, 3298562, 19395029, 114936956, 685659248, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}-4 x^{5}+7 x^{4}-8 x^{3}+6 x^{2}-3 x +1\right) F \left(x
\right)^{6}+\left(7 x^{5}-25 x^{4}+37 x^{3}-33 x^{2}+19 x -7\right) F \left(x
\right)^{5}+\left(-4 x^{5}+28 x^{4}-57 x^{3}+65 x^{2}-46 x +20\right) F \left(x
\right)^{4}+\left(x^{5}-11 x^{4}+35 x^{3}-57 x^{2}+54 x -30\right) F \left(x
\right)^{3}+\left(x^{4}-7 x^{3}+21 x^{2}-31 x +25\right) F \left(x
\right)^{2}+\left(-2 x^{2}+7 x -11\right) F \! \left(x \right)+2 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3125\)
\(\displaystyle a(8) = 17354\)
\(\displaystyle a(9) = 98386\)
\(\displaystyle a(10) = 566333\)
\(\displaystyle a(11) = 3298562\)
\(\displaystyle a(12) = 19395029\)
\(\displaystyle a(13) = 114936956\)
\(\displaystyle a(14) = 685659248\)
\(\displaystyle a(15) = 4113735616\)
\(\displaystyle a(16) = 24804490702\)
\(\displaystyle a(17) = 150223886163\)
\(\displaystyle a(18) = 913392450022\)
\(\displaystyle a(19) = 5573340531740\)
\(\displaystyle a(20) = 34116974406237\)
\(\displaystyle a(21) = 209459712729750\)
\(\displaystyle a(22) = 1289439088573261\)
\(\displaystyle a(23) = 7957570605048522\)
\(\displaystyle a(24) = 49222058346077014\)
\(\displaystyle a(25) = 305118275742300744\)
\(\displaystyle a(26) = 1895148427125600082\)
\(\displaystyle a(27) = 11793121055669711847\)
\(\displaystyle a(28) = 73514731254392163220\)
\(\displaystyle a(29) = 459023022193610999929\)
\(\displaystyle a(30) = 2870566356732461540386\)
\(\displaystyle a(31) = 17977774169692726491137\)
\(\displaystyle a(32) = 112746953679295077921706\)
\(\displaystyle a(33) = 708014967581994712255494\)
\(\displaystyle a(34) = 4451633821972072839509171\)
\(\displaystyle a(35) = 28022614948028321082543698\)
\(\displaystyle a(36) = 176597665353901346481838229\)
\(\displaystyle a(37) = 1114101995785062294476254753\)
\(\displaystyle a(38) = 7035692896468518820670978263\)
\(\displaystyle a(39) = 44474432468212033344203742288\)
\(\displaystyle a(40) = 281395137798355930171021922302\)
\(\displaystyle a(41) = 1781999683477624024925164168420\)
\(\displaystyle a(42) = 11294495780925917053678773162730\)
\(\displaystyle a(43) = 71643799754439598879852004380322\)
\(\displaystyle a(44) = 454808104863183958054180245559151\)
\(\displaystyle a(45) = 2889360570350720141113046509821074\)
\(\displaystyle a(46) = 18369028525442656778392353843897633\)
\(\displaystyle a(47) = 116860895363769291514763455572112949\)
\(\displaystyle a(48) = 743942143046264614113936700165635409\)
\(\displaystyle a(49) = 4738981745246384201664654323435529605\)
\(\displaystyle a(50) = 30206222282393890551008074560443724163\)
\(\displaystyle a(51) = 192647522153395861615635462669153151109\)
\(\displaystyle a(52) = 1229353326851322449808994332514200957199\)
\(\displaystyle a(53) = 7849236561436458235222499612567898734078\)
\(\displaystyle a(54) = 50142627979721706768947400774435946890688\)
\(\displaystyle a(55) = 320485019220183943260132887550359750168038\)
\(\displaystyle a(56) = 2049376396112511963929082677001143216386812\)
\(\displaystyle a(57) = 13111181679206990158490963298446381577105750\)
\(\displaystyle a(58) = 83919164603996018459944290823816211218494515\)
\(\displaystyle a(59) = 537369616413494248624821691754154853988331096\)
\(\displaystyle a(60) = 3442480078099806959004197936615860047966405421\)
\(\displaystyle a{\left(n + 61 \right)} = \frac{3244032 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(3 n + 1\right) \left(3 n + 2\right) a{\left(n \right)}}{\left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{135168 \left(n + 2\right) \left(n + 3\right) \left(42781 n^{3} + 262364 n^{2} + 491003 n + 289972\right) a{\left(n + 1 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{512 \left(n + 3\right) \left(97345187 n^{4} + 1048540628 n^{3} + 4109247703 n^{2} + 6926608894 n + 4229860488\right) a{\left(n + 2 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(6023 n + 338321\right) a{\left(n + 60 \right)}}{56 \left(n + 61\right)} - \frac{\left(300739 n^{2} + 33505009 n + 933161028\right) a{\left(n + 59 \right)}}{56 \left(n + 60\right) \left(n + 61\right)} + \frac{\left(9326185 n^{3} + 1545216411 n^{2} + 85335875180 n + 1570837667688\right) a{\left(n + 58 \right)}}{56 \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(67426823 n^{4} + 14765696581 n^{3} + 1212481061596 n^{2} + 44246845242000 n + 605466474043408\right) a{\left(n + 57 \right)}}{56 \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{256 \left(202149157 n^{5} + 11576184286 n^{4} + 148113071255 n^{3} + 793835503406 n^{2} + 1946449346424 n + 1807449521472\right) a{\left(n + 3 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{64 \left(4567462781 n^{5} - 1362421244806 n^{4} - 39367152990929 n^{3} - 401284933777310 n^{2} - 1769712188534376 n - 2873971503456336\right) a{\left(n + 5 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(6542235485 n^{5} + 1774979198417 n^{4} + 192609693253897 n^{3} + 10449395547581023 n^{2} + 283421459318262906 n + 3074632315834593456\right) a{\left(n + 56 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{384 \left(7111448969 n^{5} + 202822399411 n^{4} + 2197573275081 n^{3} + 11458036491653 n^{2} + 28958189959150 n + 28498017026088\right) a{\left(n + 4 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(27332637923 n^{5} + 7288306741199 n^{4} + 777311672726147 n^{3} + 41447434588368829 n^{2} + 1104929316138617294 n + 11781380028130053216\right) a{\left(n + 55 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(816435228707 n^{5} + 213919911567110 n^{4} + 22418684887332085 n^{3} + 1174651531896575422 n^{2} + 30771488190426107388 n + 322417806186912375600\right) a{\left(n + 54 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{192 \left(1160217411795 n^{5} + 48305920956134 n^{4} + 793069269991350 n^{3} + 6424576557582000 n^{2} + 25708158168532067 n + 40692123121592670\right) a{\left(n + 6 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6556960173175 n^{5} + 1687869778287868 n^{4} + 173784464057412089 n^{3} + 8946006277736752364 n^{2} + 230247168149143622568 n + 2370264709419840785472\right) a{\left(n + 53 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{96 \left(20316754508467 n^{5} + 898251302447109 n^{4} + 15789592081613712 n^{3} + 137989744643193778 n^{2} + 599744281774739094 n + 1037374818355734852\right) a{\left(n + 7 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(21399553442777 n^{5} + 5411366126542754 n^{4} + 547331728362722035 n^{3} + 27678750995308349074 n^{2} + 699834205931509144620 n + 7077621028552973017416\right) a{\left(n + 52 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(113450447441005 n^{5} + 28184582221278952 n^{4} + 2800674324135184937 n^{3} + 139145728930908785714 n^{2} + 3456480006738449803956 n + 34343594449786723099392\right) a{\left(n + 51 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(482171898043363 n^{5} + 117742205362134499 n^{4} + 11500208958820913603 n^{3} + 561607785745023198407 n^{2} + 13712443550348005703808 n + 133918767725315545261980\right) a{\left(n + 50 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1091284188975031 n^{5} + 297614247173911403 n^{4} + 31584755546637848407 n^{3} + 1642337090981023165425 n^{2} + 42038411199972748631502 n + 425124779267915459666760\right) a{\left(n + 47 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(1578147804948908 n^{5} + 379450595143142195 n^{4} + 36490405993267534666 n^{3} + 1754399982024706820959 n^{2} + 42170188917042330410688 n + 405415199440577297294460\right) a{\left(n + 49 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{24 \left(1890945100950999 n^{5} + 89826619360096255 n^{4} + 1703268325489206955 n^{3} + 16115435434059330765 n^{2} + 76083283589896853706 n + 143387930290061328120\right) a{\left(n + 8 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2322065448741637 n^{5} + 553690138320695869 n^{4} + 52783079352356203977 n^{3} + 2514637681117932540067 n^{2} + 59870869990319844371386 n + 569916758659972988741976\right) a{\left(n + 48 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{4 \left(37463083431103933 n^{5} + 1903137036705694420 n^{4} + 38697089033988351695 n^{3} + 393621470171421310820 n^{2} + 2002648784966189937972 n + 4076388353054427220080\right) a{\left(n + 9 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{6 \left(69572057380081133 n^{5} + 3785598394217053385 n^{4} + 82599551999111850145 n^{3} + 903196133464395260295 n^{2} + 4948229990813776261082 n + 10863513888463574586600\right) a{\left(n + 10 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(93011902256866884 n^{5} + 19990291681062530965 n^{4} + 1716549361313639629910 n^{3} + 73611031641973856228515 n^{2} + 1576383019317738824923686 n + 13486080981043280934486840\right) a{\left(n + 46 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(142740453104009383 n^{5} - 96507247531661019 n^{4} - 391326382901582140266 n^{3} - 12836685040283746643913 n^{2} - 158595615687107661256141 n - 698219755869709900186248\right) a{\left(n + 16 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(534646532578353984 n^{5} + 114709801269171791755 n^{4} + 9844321696842939257490 n^{3} + 422422735611970694685805 n^{2} + 9063612157758438014974646 n + 77795764763417828785297920\right) a{\left(n + 45 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(693218683979738863 n^{5} + 42970295895724061420 n^{4} + 1069758252104255565845 n^{3} + 13367716309049725597360 n^{2} + 83830820836133913190152 n + 211023915215397353980440\right) a{\left(n + 12 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(759246631185245537 n^{5} + 49850706808954071581 n^{4} + 1315053914724499901641 n^{3} + 17422575691844497685755 n^{2} + 115925177485172369725470 n + 309902137084004000167152\right) a{\left(n + 13 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(1013841752997311659 n^{5} + 59131500222632901190 n^{4} + 1384516718126015461265 n^{3} + 16263362107885178015870 n^{2} + 95817237683962871698416 n + 226451142010431969360480\right) a{\left(n + 11 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2008026729241615389 n^{5} + 138075578518375956115 n^{4} + 3809227027658288043495 n^{3} + 52706856295429974962005 n^{2} + 365804384758861876399796 n + 1018887776587093833612720\right) a{\left(n + 14 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2606787360619572427 n^{5} + 511281922180530625859 n^{4} + 40089699998591224855303 n^{3} + 1570822175038767494745993 n^{2} + 30756536098961174258582034 n + 240740107872286584362374704\right) a{\left(n + 41 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(2855573938664358907 n^{5} + 590708901056164699276 n^{4} + 48887558357902432427729 n^{3} + 2023452340528852209581840 n^{2} + 41886454938691208648817768 n + 346934592160962738347248992\right) a{\left(n + 43 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(5366323997138996113 n^{5} + 1585621681239771216625 n^{4} + 153704449527551121848621 n^{3} + 6793724667077996292204761 n^{2} + 142355923658291870297523618 n + 1152654089467082650968069018\right) a{\left(n + 34 \right)}}{28 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(5570560507424285861 n^{5} + 1175873689601563404230 n^{4} + 99301319628779982466015 n^{3} + 4193796805191331364372290 n^{2} + 88579745048327772627398124 n + 748590113025095600508316200\right) a{\left(n + 44 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6066053748609241231 n^{5} + 1109467191531978194758 n^{4} + 80607992453834109053909 n^{3} + 2904134594237145826347686 n^{2} + 51790646224842689188190184 n + 364849942225843744187867712\right) a{\left(n + 40 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6424886614563620197 n^{5} + 423996147821662443325 n^{4} + 10915631739897377075795 n^{3} + 135384728805951088768655 n^{2} + 791155288297393607913648 n + 1660072366324985073462000\right) a{\left(n + 15 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(8774215922668546863 n^{5} + 991302343602632230580 n^{4} + 42397100956741426075035 n^{3} + 875231642941332086923510 n^{2} + 8816633443589614450340352 n + 34901484578612163015199440\right) a{\left(n + 17 \right)}}{70 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(9271057637818180713 n^{5} + 1873693690173574039145 n^{4} + 151480205313022804955925 n^{3} + 6123840888321724097444835 n^{2} + 123798831642630765496634582 n + 1001241894100432158330919920\right) a{\left(n + 42 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(13959125856373083397 n^{5} - 10976297241462517399190 n^{4} - 1696046565074757204581305 n^{3} - 91532942580459961957399630 n^{2} - 2161463688568726633108618032 n - 19036942589452465348627890600\right) a{\left(n + 35 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(17837085491708728907 n^{5} + 4443466999115133986360 n^{4} + 421128839634691208513425 n^{3} + 19306944550901233025140660 n^{2} + 432329091814105866573575808 n + 3805553847596238315142781760\right) a{\left(n + 39 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(30684165375606162561 n^{5} + 8769680227458696301090 n^{4} + 860748928832017101656775 n^{3} + 39113305212417941434891550 n^{2} + 848491380719476188893310624 n + 7139619540747208228762131480\right) a{\left(n + 36 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(32313638638316388663 n^{5} + 6909470244978582313655 n^{4} + 582780913192768894228015 n^{3} + 24311129683553000755094125 n^{2} + 502687271703840604640391662 n + 4128394081141580361634522560\right) a{\left(n + 38 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(67499376199238371971 n^{5} - 581609585451240005675 n^{4} - 523776110894868142838265 n^{3} - 26588834698643188335424785 n^{2} - 515601567844458834350525246 n - 3569645236884940156824773040\right) a{\left(n + 25 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(145757944713491138873 n^{5} + 32271232050042733849775 n^{4} + 2773109064381073896204085 n^{3} + 116620300304769862933679485 n^{2} + 2413372408943385585965818302 n + 19733853385824875258374864800\right) a{\left(n + 37 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(379080983515276852933 n^{5} + 39158847891792463101235 n^{4} + 1595352730541006388449585 n^{3} + 32143831054017972751653485 n^{2} + 321017880351847364144779362 n + 1273380185867396262886325880\right) a{\left(n + 18 \right)}}{280 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(383723720583143606728 n^{5} + 59084527687790540588475 n^{4} + 3641270336920314269202786 n^{3} + 112262358739013535587748333 n^{2} + 1731366563627636950618301606 n + 10685188658356647421567237464\right) a{\left(n + 30 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(417345770056089426260 n^{5} + 62892681614651686874089 n^{4} + 3793391341434080712541386 n^{3} + 114455374771069490873171327 n^{2} + 1727355013641774615218923450 n + 10430783114580988803023780160\right) a{\left(n + 29 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(422797678885944023722 n^{5} + 74748334748345972127355 n^{4} + 5272377678034235338023470 n^{3} + 185523116375217696165720575 n^{2} + 3257604234326038024873847118 n + 22840429893006286002108622140\right) a{\left(n + 33 \right)}}{280 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(469541602276852974719 n^{5} + 76641561943999388468891 n^{4} + 5008364069761471574668471 n^{3} + 163776283487235695337672937 n^{2} + 2679818736478227887474993430 n + 17552051684808546978341521272\right) a{\left(n + 32 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(529169700627688282053 n^{5} + 54501882224255738308370 n^{4} + 2230129474287349486188415 n^{3} + 45364316437762418589475810 n^{2} + 459103583074201292659758392 n + 1850445415238228584687413120\right) a{\left(n + 19 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(716661322512404481196 n^{5} + 78999047236498874148355 n^{4} + 3429167848467747201436410 n^{3} + 72915352623086259873385765 n^{2} + 753783537118564323626431714 n + 2992895346732730918034299680\right) a{\left(n + 24 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(787069896753116145160 n^{5} + 85814347822910968000789 n^{4} + 3732627784986230178040370 n^{3} + 80949554590021529119277027 n^{2} + 875105705068828518121613502 n + 3771509573410058100879302112\right) a{\left(n + 22 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(1163819863607620034154 n^{5} + 129385472831070068189105 n^{4} + 5728758459982505247147920 n^{3} + 126193871230279988230002875 n^{2} + 1381859626924111848374797826 n + 6011406674668954180004622360\right) a{\left(n + 23 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1337650292700272668161 n^{5} + 194102912488716011537330 n^{4} + 11277016436471266802798155 n^{3} + 327718274805075960446371190 n^{2} + 4761782962648531940270765964 n + 27666558252347653444894780920\right) a{\left(n + 27 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1413040119118650146583 n^{5} + 222902853388955078880655 n^{4} + 14075106252035620725240275 n^{3} + 444677366312212067656038905 n^{2} + 7028627643007883569962683782 n + 44462657005535639992163267280\right) a{\left(n + 31 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(1943489397171213444113 n^{5} + 286509826625269840596275 n^{4} + 16838326812139527987824305 n^{3} + 492937221154879434183668845 n^{2} + 7187032035789990594085653462 n + 41750833709188770425284719600\right) a{\left(n + 26 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(2650147584463405431043 n^{5} + 277050991410122796668350 n^{4} + 11540067562183821295111385 n^{3} + 239494326691098272880280430 n^{2} + 2477132962872744483027272592 n + 10217791037227646139770468520\right) a{\left(n + 20 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(3580599563103981966976 n^{5} + 382126414482740759608705 n^{4} + 16270185805266983612898650 n^{3} + 345512982967959818361444635 n^{2} + 3659687125343268482147770674 n + 15467617546146293057883687600\right) a{\left(n + 21 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(5601762295111622101693 n^{5} + 826584056460280697358700 n^{4} + 48828252990035099103389135 n^{3} + 1443011321549103039720596000 n^{2} + 21329870209497312563089991592 n + 126136694971320561163599214680\right) a{\left(n + 28 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)}, \quad n \geq 61\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3125\)
\(\displaystyle a(8) = 17354\)
\(\displaystyle a(9) = 98386\)
\(\displaystyle a(10) = 566333\)
\(\displaystyle a(11) = 3298562\)
\(\displaystyle a(12) = 19395029\)
\(\displaystyle a(13) = 114936956\)
\(\displaystyle a(14) = 685659248\)
\(\displaystyle a(15) = 4113735616\)
\(\displaystyle a(16) = 24804490702\)
\(\displaystyle a(17) = 150223886163\)
\(\displaystyle a(18) = 913392450022\)
\(\displaystyle a(19) = 5573340531740\)
\(\displaystyle a(20) = 34116974406237\)
\(\displaystyle a(21) = 209459712729750\)
\(\displaystyle a(22) = 1289439088573261\)
\(\displaystyle a(23) = 7957570605048522\)
\(\displaystyle a(24) = 49222058346077014\)
\(\displaystyle a(25) = 305118275742300744\)
\(\displaystyle a(26) = 1895148427125600082\)
\(\displaystyle a(27) = 11793121055669711847\)
\(\displaystyle a(28) = 73514731254392163220\)
\(\displaystyle a(29) = 459023022193610999929\)
\(\displaystyle a(30) = 2870566356732461540386\)
\(\displaystyle a(31) = 17977774169692726491137\)
\(\displaystyle a(32) = 112746953679295077921706\)
\(\displaystyle a(33) = 708014967581994712255494\)
\(\displaystyle a(34) = 4451633821972072839509171\)
\(\displaystyle a(35) = 28022614948028321082543698\)
\(\displaystyle a(36) = 176597665353901346481838229\)
\(\displaystyle a(37) = 1114101995785062294476254753\)
\(\displaystyle a(38) = 7035692896468518820670978263\)
\(\displaystyle a(39) = 44474432468212033344203742288\)
\(\displaystyle a(40) = 281395137798355930171021922302\)
\(\displaystyle a(41) = 1781999683477624024925164168420\)
\(\displaystyle a(42) = 11294495780925917053678773162730\)
\(\displaystyle a(43) = 71643799754439598879852004380322\)
\(\displaystyle a(44) = 454808104863183958054180245559151\)
\(\displaystyle a(45) = 2889360570350720141113046509821074\)
\(\displaystyle a(46) = 18369028525442656778392353843897633\)
\(\displaystyle a(47) = 116860895363769291514763455572112949\)
\(\displaystyle a(48) = 743942143046264614113936700165635409\)
\(\displaystyle a(49) = 4738981745246384201664654323435529605\)
\(\displaystyle a(50) = 30206222282393890551008074560443724163\)
\(\displaystyle a(51) = 192647522153395861615635462669153151109\)
\(\displaystyle a(52) = 1229353326851322449808994332514200957199\)
\(\displaystyle a(53) = 7849236561436458235222499612567898734078\)
\(\displaystyle a(54) = 50142627979721706768947400774435946890688\)
\(\displaystyle a(55) = 320485019220183943260132887550359750168038\)
\(\displaystyle a(56) = 2049376396112511963929082677001143216386812\)
\(\displaystyle a(57) = 13111181679206990158490963298446381577105750\)
\(\displaystyle a(58) = 83919164603996018459944290823816211218494515\)
\(\displaystyle a(59) = 537369616413494248624821691754154853988331096\)
\(\displaystyle a(60) = 3442480078099806959004197936615860047966405421\)
\(\displaystyle a{\left(n + 61 \right)} = \frac{3244032 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(3 n + 1\right) \left(3 n + 2\right) a{\left(n \right)}}{\left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{135168 \left(n + 2\right) \left(n + 3\right) \left(42781 n^{3} + 262364 n^{2} + 491003 n + 289972\right) a{\left(n + 1 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{512 \left(n + 3\right) \left(97345187 n^{4} + 1048540628 n^{3} + 4109247703 n^{2} + 6926608894 n + 4229860488\right) a{\left(n + 2 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(6023 n + 338321\right) a{\left(n + 60 \right)}}{56 \left(n + 61\right)} - \frac{\left(300739 n^{2} + 33505009 n + 933161028\right) a{\left(n + 59 \right)}}{56 \left(n + 60\right) \left(n + 61\right)} + \frac{\left(9326185 n^{3} + 1545216411 n^{2} + 85335875180 n + 1570837667688\right) a{\left(n + 58 \right)}}{56 \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(67426823 n^{4} + 14765696581 n^{3} + 1212481061596 n^{2} + 44246845242000 n + 605466474043408\right) a{\left(n + 57 \right)}}{56 \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{256 \left(202149157 n^{5} + 11576184286 n^{4} + 148113071255 n^{3} + 793835503406 n^{2} + 1946449346424 n + 1807449521472\right) a{\left(n + 3 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{64 \left(4567462781 n^{5} - 1362421244806 n^{4} - 39367152990929 n^{3} - 401284933777310 n^{2} - 1769712188534376 n - 2873971503456336\right) a{\left(n + 5 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(6542235485 n^{5} + 1774979198417 n^{4} + 192609693253897 n^{3} + 10449395547581023 n^{2} + 283421459318262906 n + 3074632315834593456\right) a{\left(n + 56 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{384 \left(7111448969 n^{5} + 202822399411 n^{4} + 2197573275081 n^{3} + 11458036491653 n^{2} + 28958189959150 n + 28498017026088\right) a{\left(n + 4 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(27332637923 n^{5} + 7288306741199 n^{4} + 777311672726147 n^{3} + 41447434588368829 n^{2} + 1104929316138617294 n + 11781380028130053216\right) a{\left(n + 55 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(816435228707 n^{5} + 213919911567110 n^{4} + 22418684887332085 n^{3} + 1174651531896575422 n^{2} + 30771488190426107388 n + 322417806186912375600\right) a{\left(n + 54 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{192 \left(1160217411795 n^{5} + 48305920956134 n^{4} + 793069269991350 n^{3} + 6424576557582000 n^{2} + 25708158168532067 n + 40692123121592670\right) a{\left(n + 6 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6556960173175 n^{5} + 1687869778287868 n^{4} + 173784464057412089 n^{3} + 8946006277736752364 n^{2} + 230247168149143622568 n + 2370264709419840785472\right) a{\left(n + 53 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{96 \left(20316754508467 n^{5} + 898251302447109 n^{4} + 15789592081613712 n^{3} + 137989744643193778 n^{2} + 599744281774739094 n + 1037374818355734852\right) a{\left(n + 7 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(21399553442777 n^{5} + 5411366126542754 n^{4} + 547331728362722035 n^{3} + 27678750995308349074 n^{2} + 699834205931509144620 n + 7077621028552973017416\right) a{\left(n + 52 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(113450447441005 n^{5} + 28184582221278952 n^{4} + 2800674324135184937 n^{3} + 139145728930908785714 n^{2} + 3456480006738449803956 n + 34343594449786723099392\right) a{\left(n + 51 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(482171898043363 n^{5} + 117742205362134499 n^{4} + 11500208958820913603 n^{3} + 561607785745023198407 n^{2} + 13712443550348005703808 n + 133918767725315545261980\right) a{\left(n + 50 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1091284188975031 n^{5} + 297614247173911403 n^{4} + 31584755546637848407 n^{3} + 1642337090981023165425 n^{2} + 42038411199972748631502 n + 425124779267915459666760\right) a{\left(n + 47 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(1578147804948908 n^{5} + 379450595143142195 n^{4} + 36490405993267534666 n^{3} + 1754399982024706820959 n^{2} + 42170188917042330410688 n + 405415199440577297294460\right) a{\left(n + 49 \right)}}{56 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{24 \left(1890945100950999 n^{5} + 89826619360096255 n^{4} + 1703268325489206955 n^{3} + 16115435434059330765 n^{2} + 76083283589896853706 n + 143387930290061328120\right) a{\left(n + 8 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2322065448741637 n^{5} + 553690138320695869 n^{4} + 52783079352356203977 n^{3} + 2514637681117932540067 n^{2} + 59870869990319844371386 n + 569916758659972988741976\right) a{\left(n + 48 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{4 \left(37463083431103933 n^{5} + 1903137036705694420 n^{4} + 38697089033988351695 n^{3} + 393621470171421310820 n^{2} + 2002648784966189937972 n + 4076388353054427220080\right) a{\left(n + 9 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{6 \left(69572057380081133 n^{5} + 3785598394217053385 n^{4} + 82599551999111850145 n^{3} + 903196133464395260295 n^{2} + 4948229990813776261082 n + 10863513888463574586600\right) a{\left(n + 10 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(93011902256866884 n^{5} + 19990291681062530965 n^{4} + 1716549361313639629910 n^{3} + 73611031641973856228515 n^{2} + 1576383019317738824923686 n + 13486080981043280934486840\right) a{\left(n + 46 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(142740453104009383 n^{5} - 96507247531661019 n^{4} - 391326382901582140266 n^{3} - 12836685040283746643913 n^{2} - 158595615687107661256141 n - 698219755869709900186248\right) a{\left(n + 16 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(534646532578353984 n^{5} + 114709801269171791755 n^{4} + 9844321696842939257490 n^{3} + 422422735611970694685805 n^{2} + 9063612157758438014974646 n + 77795764763417828785297920\right) a{\left(n + 45 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(693218683979738863 n^{5} + 42970295895724061420 n^{4} + 1069758252104255565845 n^{3} + 13367716309049725597360 n^{2} + 83830820836133913190152 n + 211023915215397353980440\right) a{\left(n + 12 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(759246631185245537 n^{5} + 49850706808954071581 n^{4} + 1315053914724499901641 n^{3} + 17422575691844497685755 n^{2} + 115925177485172369725470 n + 309902137084004000167152\right) a{\left(n + 13 \right)}}{7 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(1013841752997311659 n^{5} + 59131500222632901190 n^{4} + 1384516718126015461265 n^{3} + 16263362107885178015870 n^{2} + 95817237683962871698416 n + 226451142010431969360480\right) a{\left(n + 11 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2008026729241615389 n^{5} + 138075578518375956115 n^{4} + 3809227027658288043495 n^{3} + 52706856295429974962005 n^{2} + 365804384758861876399796 n + 1018887776587093833612720\right) a{\left(n + 14 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(2606787360619572427 n^{5} + 511281922180530625859 n^{4} + 40089699998591224855303 n^{3} + 1570822175038767494745993 n^{2} + 30756536098961174258582034 n + 240740107872286584362374704\right) a{\left(n + 41 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(2855573938664358907 n^{5} + 590708901056164699276 n^{4} + 48887558357902432427729 n^{3} + 2023452340528852209581840 n^{2} + 41886454938691208648817768 n + 346934592160962738347248992\right) a{\left(n + 43 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(5366323997138996113 n^{5} + 1585621681239771216625 n^{4} + 153704449527551121848621 n^{3} + 6793724667077996292204761 n^{2} + 142355923658291870297523618 n + 1152654089467082650968069018\right) a{\left(n + 34 \right)}}{28 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(5570560507424285861 n^{5} + 1175873689601563404230 n^{4} + 99301319628779982466015 n^{3} + 4193796805191331364372290 n^{2} + 88579745048327772627398124 n + 748590113025095600508316200\right) a{\left(n + 44 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6066053748609241231 n^{5} + 1109467191531978194758 n^{4} + 80607992453834109053909 n^{3} + 2904134594237145826347686 n^{2} + 51790646224842689188190184 n + 364849942225843744187867712\right) a{\left(n + 40 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(6424886614563620197 n^{5} + 423996147821662443325 n^{4} + 10915631739897377075795 n^{3} + 135384728805951088768655 n^{2} + 791155288297393607913648 n + 1660072366324985073462000\right) a{\left(n + 15 \right)}}{35 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(8774215922668546863 n^{5} + 991302343602632230580 n^{4} + 42397100956741426075035 n^{3} + 875231642941332086923510 n^{2} + 8816633443589614450340352 n + 34901484578612163015199440\right) a{\left(n + 17 \right)}}{70 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(9271057637818180713 n^{5} + 1873693690173574039145 n^{4} + 151480205313022804955925 n^{3} + 6123840888321724097444835 n^{2} + 123798831642630765496634582 n + 1001241894100432158330919920\right) a{\left(n + 42 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(13959125856373083397 n^{5} - 10976297241462517399190 n^{4} - 1696046565074757204581305 n^{3} - 91532942580459961957399630 n^{2} - 2161463688568726633108618032 n - 19036942589452465348627890600\right) a{\left(n + 35 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(17837085491708728907 n^{5} + 4443466999115133986360 n^{4} + 421128839634691208513425 n^{3} + 19306944550901233025140660 n^{2} + 432329091814105866573575808 n + 3805553847596238315142781760\right) a{\left(n + 39 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(30684165375606162561 n^{5} + 8769680227458696301090 n^{4} + 860748928832017101656775 n^{3} + 39113305212417941434891550 n^{2} + 848491380719476188893310624 n + 7139619540747208228762131480\right) a{\left(n + 36 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(32313638638316388663 n^{5} + 6909470244978582313655 n^{4} + 582780913192768894228015 n^{3} + 24311129683553000755094125 n^{2} + 502687271703840604640391662 n + 4128394081141580361634522560\right) a{\left(n + 38 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(67499376199238371971 n^{5} - 581609585451240005675 n^{4} - 523776110894868142838265 n^{3} - 26588834698643188335424785 n^{2} - 515601567844458834350525246 n - 3569645236884940156824773040\right) a{\left(n + 25 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(145757944713491138873 n^{5} + 32271232050042733849775 n^{4} + 2773109064381073896204085 n^{3} + 116620300304769862933679485 n^{2} + 2413372408943385585965818302 n + 19733853385824875258374864800\right) a{\left(n + 37 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(379080983515276852933 n^{5} + 39158847891792463101235 n^{4} + 1595352730541006388449585 n^{3} + 32143831054017972751653485 n^{2} + 321017880351847364144779362 n + 1273380185867396262886325880\right) a{\left(n + 18 \right)}}{280 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(383723720583143606728 n^{5} + 59084527687790540588475 n^{4} + 3641270336920314269202786 n^{3} + 112262358739013535587748333 n^{2} + 1731366563627636950618301606 n + 10685188658356647421567237464\right) a{\left(n + 30 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(417345770056089426260 n^{5} + 62892681614651686874089 n^{4} + 3793391341434080712541386 n^{3} + 114455374771069490873171327 n^{2} + 1727355013641774615218923450 n + 10430783114580988803023780160\right) a{\left(n + 29 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(422797678885944023722 n^{5} + 74748334748345972127355 n^{4} + 5272377678034235338023470 n^{3} + 185523116375217696165720575 n^{2} + 3257604234326038024873847118 n + 22840429893006286002108622140\right) a{\left(n + 33 \right)}}{280 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(469541602276852974719 n^{5} + 76641561943999388468891 n^{4} + 5008364069761471574668471 n^{3} + 163776283487235695337672937 n^{2} + 2679818736478227887474993430 n + 17552051684808546978341521272\right) a{\left(n + 32 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(529169700627688282053 n^{5} + 54501882224255738308370 n^{4} + 2230129474287349486188415 n^{3} + 45364316437762418589475810 n^{2} + 459103583074201292659758392 n + 1850445415238228584687413120\right) a{\left(n + 19 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(716661322512404481196 n^{5} + 78999047236498874148355 n^{4} + 3429167848467747201436410 n^{3} + 72915352623086259873385765 n^{2} + 753783537118564323626431714 n + 2992895346732730918034299680\right) a{\left(n + 24 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(787069896753116145160 n^{5} + 85814347822910968000789 n^{4} + 3732627784986230178040370 n^{3} + 80949554590021529119277027 n^{2} + 875105705068828518121613502 n + 3771509573410058100879302112\right) a{\left(n + 22 \right)}}{112 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{3 \left(1163819863607620034154 n^{5} + 129385472831070068189105 n^{4} + 5728758459982505247147920 n^{3} + 126193871230279988230002875 n^{2} + 1381859626924111848374797826 n + 6011406674668954180004622360\right) a{\left(n + 23 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1337650292700272668161 n^{5} + 194102912488716011537330 n^{4} + 11277016436471266802798155 n^{3} + 327718274805075960446371190 n^{2} + 4761782962648531940270765964 n + 27666558252347653444894780920\right) a{\left(n + 27 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{3 \left(1413040119118650146583 n^{5} + 222902853388955078880655 n^{4} + 14075106252035620725240275 n^{3} + 444677366312212067656038905 n^{2} + 7028627643007883569962683782 n + 44462657005535639992163267280\right) a{\left(n + 31 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(1943489397171213444113 n^{5} + 286509826625269840596275 n^{4} + 16838326812139527987824305 n^{3} + 492937221154879434183668845 n^{2} + 7187032035789990594085653462 n + 41750833709188770425284719600\right) a{\left(n + 26 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} - \frac{\left(2650147584463405431043 n^{5} + 277050991410122796668350 n^{4} + 11540067562183821295111385 n^{3} + 239494326691098272880280430 n^{2} + 2477132962872744483027272592 n + 10217791037227646139770468520\right) a{\left(n + 20 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(3580599563103981966976 n^{5} + 382126414482740759608705 n^{4} + 16270185805266983612898650 n^{3} + 345512982967959818361444635 n^{2} + 3659687125343268482147770674 n + 15467617546146293057883687600\right) a{\left(n + 21 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)} + \frac{\left(5601762295111622101693 n^{5} + 826584056460280697358700 n^{4} + 48828252990035099103389135 n^{3} + 1443011321549103039720596000 n^{2} + 21329870209497312563089991592 n + 126136694971320561163599214680\right) a{\left(n + 28 \right)}}{560 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right) \left(n + 61\right)}, \quad n \geq 61\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 63 rules.
Finding the specification took 6849 seconds.
Copy 63 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_{23}\! \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_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{38}\! \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_{18}\! \left(x \right) F_{23}\! \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_{27}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{32}\! \left(x \right) x +F_{32} \left(x \right)^{2}-2 F_{32}\! \left(x \right)+2\\
F_{33}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{32}\! \left(x \right) F_{45}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{23}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right) F_{32}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{14}\! \left(x \right) F_{23}\! \left(x \right) F_{32}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{60}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{21}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{23}\! \left(x \right) F_{32}\! \left(x \right) F_{55}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 63 rules.
Finding the specification took 17233 seconds.
Copy 63 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_{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_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{19}\! \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) &= \frac{F_{22}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{2}\! \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_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{30}\! \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_{36}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{36}\! \left(x \right) x +F_{36} \left(x \right)^{2}+x\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{45}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{44}\! \left(x \right) x +F_{44} \left(x \right)^{2}-2 F_{44}\! \left(x \right)+2\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{26}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{24}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{18}\! \left(x \right) F_{26}\! \left(x \right) F_{44}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{17}\! \left(x \right) F_{26}\! \left(x \right) F_{44}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{60}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{26}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{24}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{26}\! \left(x \right) F_{44}\! \left(x \right) F_{55}\! \left(x \right)\\
\end{align*}\)