1, 1, 2, 6, 24, 114, 596, 3290, 18790, 109890, 654306, 3952002, 24153210, 149089166, 928125902, ...

\(\displaystyle 2 x^{3} \left(x^{2}-5 x +3\right) F \left(x \right)^{5}+x \left(2 x^{3}+2 x^{2}-6 x -1\right) F \left(x \right)^{4}+\left(-6 x^{3}+9 x^{2}+6 x +1\right) F \left(x \right)^{3}+\left(-x^{2}-10 x -5\right) F \left(x \right)^{2}+\left(4 x +8\right) F \! \left(x \right)-4 = 0\)

\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a(6) = 596\)
\(\displaystyle a(7) = 3290\)
\(\displaystyle a(8) = 18790\)
\(\displaystyle a(9) = 109890\)
\(\displaystyle a(10) = 654306\)
\(\displaystyle a(11) = 3952002\)
\(\displaystyle a(12) = 24153210\)
\(\displaystyle a(13) = 149089166\)
\(\displaystyle a(14) = 928125902\)
\(\displaystyle a(15) = 5820517922\)
\(\displaystyle a(16) = 36737421498\)
\(\displaystyle a(17) = 233194122382\)
\(\displaystyle a(18) = 1487692594630\)
\(\displaystyle a(19) = 9533750805570\)
\(\displaystyle a(20) = 61344062686442\)
\(\displaystyle a(21) = 396160133264566\)
\(\displaystyle a(22) = 2566923502650190\)
\(\displaystyle a(23) = 16682936569283754\)
\(\displaystyle a(24) = 108727364670610386\)
\(\displaystyle a(25) = 710419267423782798\)
\(\displaystyle a(26) = 4652797018683778230\)
\(\displaystyle a(27) = 30539406508605286386\)
\(\displaystyle a(28) = 200856563122914397322\)
\(\displaystyle a(29) = 1323516432052793083526\)
\(\displaystyle a(30) = 8736462055781510699550\)
\(\displaystyle a(31) = 57763728335619763038938\)
\(\displaystyle a(32) = 382510442747554793411778\)
\(\displaystyle a(33) = 2536641144555125995716926\)
\(\displaystyle a(34) = 16844776246035434649671878\)
\(\displaystyle a(35) = 112002563866929196012346818\)
\(\displaystyle a(36) = 745617443215174486672163386\)
\(\displaystyle a(37) = 4969365540509788674965756726\)
\(\displaystyle a(38) = 33155554835482753872021427438\)
\(\displaystyle a(39) = 221440903508422189738662710442\)
\(\displaystyle a(40) = 1480413809981031225019586455922\)
\(\displaystyle a(41) = 9906297125270374996865305874286\)
\(\displaystyle a(42) = 66347292672565193730071448461078\)
\(\displaystyle a(43) = 444734422384407684227749979989394\)
\(\displaystyle a(44) = 2983508274142227420239836723909354\)
\(\displaystyle a(45) = 20030284169061967710872161080715494\)
\(\displaystyle a(46) = 134575425003042380878044779554211006\)
\(\displaystyle a(47) = 904793668379106346400568970232434810\)
\(\displaystyle a(48) = 6087313993839429010962886306597338786\)
\(\displaystyle a(49) = 40980968258629099599582226433197960478\)
\(\displaystyle a(50) = 276062751289055088752648946699748149350\)
\(\displaystyle a(51) = 1860766529498200374059602945631768420834\)
\(\displaystyle a(52) = 12549442786604730247300961605679270385946\)
\(\displaystyle a(53) = 84682965676194106654971795922397075740054\)
\(\displaystyle a(54) = 571739022616583550893919770015526903174798\)
\(\displaystyle a(55) = 3862080834153240405178949054828762884824650\)
\(\displaystyle a(56) = 26101091791366934707269742005468672532637138\)
\(\displaystyle a(57) = 176482741208544316868777320379503119345948110\)
\(\displaystyle a{\left(n + 58 \right)} = \frac{927532453154902016 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(n + 4\right) a{\left(n \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{64 \left(n + 2\right) \left(n + 3\right) \left(n + 4\right) \left(123818086691602921 n + 1197693860636093162\right) a{\left(n + 1 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{4 \left(n + 3\right) \left(n + 4\right) \left(5519953681636569866 n^{2} + 24891835296394931735 n - 73815275510403679502\right) a{\left(n + 2 \right)}}{27 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{4 \left(n + 4\right) \left(224250341559453600117 n^{3} + 4708378953467647242089 n^{2} + 31936504094172927519009 n + 69751547441849936755559\right) a{\left(n + 3 \right)}}{27 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(869 n + 48580\right) a{\left(n + 57 \right)}}{6 \left(n + 60\right)} - \frac{\left(714875 n^{2} + 79189689 n + 2192762920\right) a{\left(n + 56 \right)}}{72 \left(n + 59\right) \left(n + 60\right)} + \frac{\left(184494785 n^{3} + 30367869921 n^{2} + 1665904895038 n + 30457271389200\right) a{\left(n + 55 \right)}}{432 \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(33357627992 n^{4} + 7250450765781 n^{3} + 590844218435881 n^{2} + 21394607868826818 n + 290450274310021680\right) a{\left(n + 54 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(742999699491 n^{4} + 158249761363636 n^{3} + 12635910669694152 n^{2} + 448294249378408799 n + 5962446366003748170\right) a{\left(n + 53 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(12515322451456 n^{4} + 2608047152443360 n^{3} + 203723950332345281 n^{2} + 7069736699951981453 n + 91961791965732657114\right) a{\left(n + 52 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(26406520046042 n^{4} + 5367883029674229 n^{3} + 408901429518597607 n^{2} + 13833440926929780341 n + 175362431484243662031\right) a{\left(n + 51 \right)}}{432 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(1415345761430065 n^{4} + 278369027224182654 n^{3} + 20495871410622951584 n^{2} + 669459240157381773777 n + 8183491714151172997296\right) a{\left(n + 50 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(6398437846768561 n^{4} + 1157649276632360930 n^{3} + 77574625929548261738 n^{2} + 2274356619129715846795 n + 24497201257783770021384\right) a{\left(n + 49 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(46729594831121769 n^{4} + 10395194241861015989 n^{3} + 856137177893389741104 n^{2} + 31015459610697268169374 n + 417769380178256525142588\right) a{\left(n + 48 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(700839160696791815 n^{4} + 139985247530928862863 n^{3} + 10476497649498905376190 n^{2} + 348195430366800857489712 n + 4336431727508018642741148\right) a{\left(n + 47 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(17751119893363137107 n^{4} + 3419924720884394645314 n^{3} + 247079140531533262727773 n^{2} + 7933538613317220040843802 n + 95525513239494641799036036\right) a{\left(n + 46 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(77515956294854942529 n^{4} + 14536220523062321371400 n^{3} + 1022505589650918891798135 n^{2} + 31975218909037475459918170 n + 375061415004638277525795240\right) a{\left(n + 45 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(989579854297313932159 n^{4} + 181221050396866821807380 n^{3} + 12451954196610383943729635 n^{2} + 380464591296518914151368582 n + 4361569925700171643906433448\right) a{\left(n + 44 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(2103310248447649369936 n^{4} + 377506712953487355287746 n^{3} + 25436644618715334313138361 n^{2} + 762556807971440749884216143 n + 8581272060978114675024123648\right) a{\left(n + 43 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(4133600183236680682037 n^{4} + 764955584241430914540628 n^{3} + 53305494489700049055431275 n^{2} + 1655883739245297393694722440 n + 19328420382643131217543418136\right) a{\left(n + 42 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{4 \left(9111582252460201039817 n^{4} + 247885154543540858993340 n^{3} + 2495415816591072548048677 n^{2} + 10992786253567528216323252 n + 17869392395877832174827510\right) a{\left(n + 4 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{4 \left(12257892145724816806947 n^{4} + 402789594847815030348306 n^{3} + 4852029062514812191190807 n^{2} + 25427203377517231803481296 n + 49006189660848223197866248\right) a{\left(n + 5 \right)}}{27 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(60884222829911811559494 n^{4} + 10169447526952494923941400 n^{3} + 635570509007842430860935453 n^{2} + 17613779744725151268823630201 n + 182613468702027797527421396952\right) a{\left(n + 41 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{4 \left(113605607335098351873483 n^{4} + 2359790250299403525295739 n^{3} + 13583928798799944695349336 n^{2} - 1828958861141872760672165 n - 141766544517167696646899100\right) a{\left(n + 6 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(1412052172477499106009637 n^{4} + 231803661071980318833647144 n^{3} + 14255857226289660790280655029 n^{2} + 389269152093106172573282534638 n + 3981960155168767565997228087192\right) a{\left(n + 40 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(2549610479098150243265162 n^{4} + 409024719161090836483496103 n^{3} + 24590818093699353294645656818 n^{2} + 656641533044068352221109920425 n + 6570934779729758503279180317846\right) a{\left(n + 39 \right)}}{648 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{2 \left(4239399789391902970385666 n^{4} + 134764780887054957969608447 n^{3} + 1594215986200242613895240743 n^{2} + 8311670997049565114000799268 n + 16098619732263148241692226070\right) a{\left(n + 7 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{2 \left(25755174003628831295500177 n^{4} + 935752631666118806647496012 n^{3} + 12731797431709452737430171782 n^{2} + 76887360880550029026582382111 n + 173894984297333636380190516658\right) a{\left(n + 8 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(57416932092838636810923285 n^{4} + 8986808100385615844094259991 n^{3} + 527213035182132639237026659392 n^{2} + 13739392713116622508221430546324 n + 134203127298023683511237821677144\right) a{\left(n + 38 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{2 \left(99326532895595584016844475 n^{4} + 4009973928155994897513837325 n^{3} + 60714986818714701733917264971 n^{2} + 408636855932791783858359153581 n + 1031579377806375601634658316578\right) a{\left(n + 9 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(182613498909160401799394616 n^{4} + 8094497543032952177518823259 n^{3} + 134622180927899312476308182615 n^{2} + 995664658148891380989786007234 n + 2763160192474146556052541376480\right) a{\left(n + 10 \right)}}{27 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(268885889793764307796594361 n^{4} + 41018706952891217566520103212 n^{3} + 2345581027008784607983286968978 n^{2} + 59588101798130649021520621568167 n + 567443899331846754313690150637778\right) a{\left(n + 37 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(685392146826887181311532085 n^{4} + 35729367618155416207995831938 n^{3} + 689407553971693003761605969384 n^{2} + 5815121197449192440916348833143 n + 17999241351822542277092418755256\right) a{\left(n + 13 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(1077011294887262174651409227 n^{4} + 159995553789106362059391872494 n^{3} + 8909964064098326411914116151762 n^{2} + 220450352923786778133350683800563 n + 2044685108877836974346895153184602\right) a{\left(n + 36 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(1101463714730756549504877989 n^{4} + 53127314807740111283435860314 n^{3} + 961390968796632127490267904292 n^{2} + 7735821134614239850367386022103 n + 23353605116439831326127693737664\right) a{\left(n + 11 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(1487393002002916082956456934 n^{4} + 77175275005012687493357081657 n^{3} + 1501165666158708578062363216948 n^{2} + 12973188675798072850050970178305 n + 42027115308766941833541529920090\right) a{\left(n + 12 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(1870865066174473853716619623 n^{4} + 270419362821575667797078668067 n^{3} + 14653174654863668109331573236554 n^{2} + 352785633839830228619631160669813 n + 3184120273112610467371585281832761\right) a{\left(n + 35 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(2763788226262727653848257653 n^{4} + 176003667049621809025418826107 n^{3} + 4205762982688074802777334055245 n^{2} + 44687207243537785619632833971212 n + 178106842211456264667907574950062\right) a{\left(n + 14 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(9586549291570073304156426394 n^{4} + 639610992402700407781395537380 n^{3} + 16011818332493611495554861842834 n^{2} + 178233063667890633092228795718949 n + 744285707401568044613678357276775\right) a{\left(n + 15 \right)}}{81 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(11356920349273742705898946828 n^{4} + 1595807320314335330659244877712 n^{3} + 84064413989167903388461562220881 n^{2} + 1967625701522973083403438405194795 n + 17265757365497565913881605399321208\right) a{\left(n + 34 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(11559103668884390975632555691 n^{4} - 6281921251268839243839009876 n^{3} - 31812742291414748737939884838979 n^{2} - 899834775921933955224118032785574 n - 7173310632118261308818970412588458\right) a{\left(n + 20 \right)}}{324 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(30211606258423565162718665343 n^{4} + 4122852900769244863733876781442 n^{3} + 210932279481042143576546517599448 n^{2} + 4795083227866676807824416251510135 n + 40866878717820159576296899368339312\right) a{\left(n + 33 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(36149142407736231396196989844 n^{4} + 2562163254591520713828196819065 n^{3} + 68105028490923499867920926847245 n^{2} + 804608363689815173654661634027320 n + 3564683765696112720517695438456108\right) a{\left(n + 16 \right)}}{162 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(45012310155522801478097724369 n^{4} + 4347354263863676635675425308615 n^{3} + 153440784635695078225956239171391 n^{2} + 2361360147295211466412736402950453 n + 13427754987426588682762006775377638\right) a{\left(n + 19 \right)}}{324 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(46891700708723879072571555615 n^{4} + 4443946838947474901532682308521 n^{3} + 155997026564930271447192347568139 n^{2} + 2396126538273313428002565800412925 n + 13523756383943238290692742072259442\right) a{\left(n + 25 \right)}}{216 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(47396309665665041838249847215 n^{4} + 3587884887042152592261145540322 n^{3} + 101776154766367857329173216091100 n^{2} + 1282220147647516409621822220904987 n + 6053557999593723256041079116080880\right) a{\left(n + 17 \right)}}{162 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(55690156017988828319453953787 n^{4} + 3733589184121167395711453187704 n^{3} + 83676943448771138642501784264295 n^{2} + 632333257199137445691868925010424 n + 121209057420859639831621582153854\right) a{\left(n + 21 \right)}}{324 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(70465040417298820134299356436 n^{4} + 9328633592217621318526240478613 n^{3} + 463008434938842336087049043216191 n^{2} + 10211109934514904357788091225442932 n + 84427598232831718509078939165092508\right) a{\left(n + 32 \right)}}{2592 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(71912310919011166727956115764 n^{4} + 9223090567024398174063006334549 n^{3} + 443482164065031221757330996482543 n^{2} + 9475207195668216906921845333319680 n + 75897622731132531562415162970528216\right) a{\left(n + 31 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(73465097070301815647477765877 n^{4} + 5516971081494849131753466145763 n^{3} + 146387061898093588343292705026241 n^{2} + 1553535644673413228062352163663463 n + 4859032166275867496249548372668900\right) a{\left(n + 22 \right)}}{324 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(86769891845132950578199923226 n^{4} + 7125204451716372173253780346607 n^{3} + 218655301900974473552608106606333 n^{2} + 2973039602541532121454523075789864 n + 15117028111095553287429229059022824\right) a{\left(n + 18 \right)}}{324 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(127951038788148285272075706311 n^{4} + 15869495372990335522840700766007 n^{3} + 737902158403545910906919215328302 n^{2} + 15245323098518300422636313425872350 n + 118083741069338424373063068458403750\right) a{\left(n + 30 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(136169425748288441466300559931 n^{4} + 11747538674805712094965636181632 n^{3} + 368657807875081533950783603228464 n^{2} + 4916534600570853602880825687125189 n + 22866433900507967480241282960280542\right) a{\left(n + 24 \right)}}{648 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(144238000877476568214064847570 n^{4} + 11532376813014876398657661161843 n^{3} + 329730956886156528366024600531235 n^{2} + 3874514562510703199373589403975146 n + 14632266249253476694113370433317860\right) a{\left(n + 23 \right)}}{648 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(197394696579478230185994171583 n^{4} + 23613292124916486867921495323576 n^{3} + 1058913203171209711680997477553510 n^{2} + 21097503638144055553653726208463239 n + 157572462559213724378737292956111170\right) a{\left(n + 29 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(262605950127550868857339920746 n^{4} + 30169216937536164450689727184515 n^{3} + 1298989362060859866866735299795192 n^{2} + 24843174997097404690006796024154987 n + 178062932972023194247897283659519386\right) a{\left(n + 28 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} + \frac{\left(300938216027772145588950863399 n^{4} + 32950004651805953972301163927284 n^{3} + 1351113923699370063439723692256630 n^{2} + 24588625162140813745870993215262335 n + 167553320662293025779208956071036514\right) a{\left(n + 27 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)} - \frac{\left(301872410037878974032334850204 n^{4} + 31067739463481733969467699657943 n^{3} + 1194345771202711419324762721371961 n^{2} + 20316301564491345507843322737530142 n + 128941159705310875664118634521033072\right) a{\left(n + 26 \right)}}{1296 \left(n + 57\right) \left(n + 58\right) \left(n + 59\right) \left(n + 60\right)}, \quad n \geq 58\)