Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432)
Counting Sequence
1, 1, 2, 6, 24, 108, 517, 2580, 13277, 69956, 375523, 2046393, 11291232, 62954575, 354139163, ...
Implicit Equation for the Generating Function
\(\displaystyle -x F \left(x
\right)^{6}+x \left(x +3\right) F \left(x
\right)^{5}-x \left(3 x +4\right) F \left(x
\right)^{4}+2 x \left(1+2 x \right) F \left(x
\right)^{3}-2 x^{2} F \left(x
\right)^{2}+\left(1-x \right) F \! \left(x \right)-1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 517\)
\(\displaystyle a(7) = 2580\)
\(\displaystyle a(8) = 13277\)
\(\displaystyle a(9) = 69956\)
\(\displaystyle a(10) = 375523\)
\(\displaystyle a(11) = 2046393\)
\(\displaystyle a(12) = 11291232\)
\(\displaystyle a(13) = 62954575\)
\(\displaystyle a(14) = 354139163\)
\(\displaystyle a(15) = 2007464860\)
\(\displaystyle a(16) = 11455640891\)
\(\displaystyle a(17) = 65756198379\)
\(\displaystyle a(18) = 379409903992\)
\(\displaystyle a(19) = 2199334938209\)
\(\displaystyle a(20) = 12802019416555\)
\(\displaystyle a(21) = 74798669115432\)
\(\displaystyle a(22) = 438516581229747\)
\(\displaystyle a(23) = 2578835415720416\)
\(\displaystyle a(24) = 15208702319342870\)
\(\displaystyle a(25) = 89927210626144019\)
\(\displaystyle a(26) = 533005690952120498\)
\(\displaystyle a(27) = 3166180739828123982\)
\(\displaystyle a(28) = 18846584060274483157\)
\(\displaystyle a(29) = 112398367697187844181\)
\(\displaystyle a(30) = 671523911133583255448\)
\(\displaystyle a(31) = 4018706774817784236602\)
\(\displaystyle a(32) = 24087326202516995486315\)
\(\displaystyle a(33) = 144586126453184080411950\)
\(\displaystyle a(34) = 869085401365072758359137\)
\(\displaystyle a(35) = 5230719600455432818148141\)
\(\displaystyle a(36) = 31520409831325553252519341\)
\(\displaystyle a(37) = 190162372803590296056952175\)
\(\displaystyle a(38) = 1148504895789457613160868957\)
\(\displaystyle a(39) = 6943715945734894247920438085\)
\(\displaystyle a(40) = 42022225778307662507192051567\)
\(\displaystyle a(41) = 254549961137867869415255024247\)
\(\displaystyle a(42) = 1543313984587182195685210289899\)
\(\displaystyle a(43) = 9364931397955286156486818280632\)
\(\displaystyle a(44) = 56873112577757474259849429633394\)
\(\displaystyle a(45) = 345657242683256160441523582087493\)
\(\displaystyle a(46) = 2102353851295376800979994736917069\)
\(\displaystyle a(47) = 12795982379336775491182499776854016\)
\(\displaystyle a(48) = 77935650654475916813024286490921993\)
\(\displaystyle a(49) = 474986493951181041883645820199913714\)
\(\displaystyle a(50) = 2896659886230297525360326238757884004\)
\(\displaystyle a(51) = 17675599896983326737783387011002382768\)
\(\displaystyle a(52) = 107919797467564943383654218940807259215\)
\(\displaystyle a(53) = 659278404034369402375256144570834899233\)
\(\displaystyle a(54) = 4029659479143810004063337689269488381852\)
\(\displaystyle a(55) = 24642860213918857626042963054884352530838\)
\(\displaystyle a(56) = 150774918311369904033493038993550563200134\)
\(\displaystyle a{\left(n + 57 \right)} = \frac{2193632 n \left(n - 1\right) \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{255 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{56 n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(308311 n + 1528964\right) a{\left(n + 1 \right)}}{51 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{4 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(41264566 n^{2} + 492378809 n + 1392163335\right) a{\left(n + 2 \right)}}{255 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(n + 2\right) \left(n + 3\right) \left(9304000366 n^{3} + 166265700750 n^{2} + 933626654579 n + 1670744516340\right) a{\left(n + 3 \right)}}{1785 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(n + 3\right) \left(163759458917 n^{4} + 3999140808064 n^{3} + 35759941038013 n^{2} + 138376022277866 n + 195392634704280\right) a{\left(n + 4 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(36429503 n^{5} + 10123667410 n^{4} + 1125306256805 n^{3} + 62540562021650 n^{2} + 1737844576161512 n + 19315595597188320\right) a{\left(n + 56 \right)}}{170 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(3897317477 n^{5} + 661969065250 n^{4} + 29917714105535 n^{3} - 598336751858150 n^{2} - 75090905798490312 n - 1438444165524413520\right) a{\left(n + 54 \right)}}{4760 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(55579306351 n^{5} + 15172113565880 n^{4} + 1656666951679805 n^{3} + 90446257184215840 n^{2} + 2468950325947482204 n + 26958178791399337200\right) a{\left(n + 55 \right)}}{7140 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(275620558993 n^{5} + 8683164741110 n^{4} + 109936565938495 n^{3} + 689209032103810 n^{2} + 2121777905252192 n + 2555051413977120\right) a{\left(n + 5 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(10094380043275 n^{5} + 2627468565737690 n^{4} + 273561113348769565 n^{3} + 14240958900222810806 n^{2} + 370673232619048084224 n + 3859219047263196299616\right) a{\left(n + 52 \right)}}{1904 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11048636087637 n^{5} + 2883892018243620 n^{4} + 301092030515000555 n^{3} + 15717373612054815120 n^{2} + 410224565435420680588 n + 4282672360040962697520\right) a{\left(n + 53 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11905948444723 n^{5} + 496318654712570 n^{4} + 8377694457189115 n^{3} + 71152983544547710 n^{2} + 302658214512471562 n + 514109679835712340\right) a{\left(n + 8 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(13672278486428 n^{5} + 686160433491055 n^{4} + 12993775697642550 n^{3} + 117764901650282545 n^{2} + 515490304286908022 n + 877120393864696320\right) a{\left(n + 7 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(20510953891186 n^{5} + 796930701558185 n^{4} + 12208339786699580 n^{3} + 91760290676249515 n^{2} + 337755239332506774 n + 486827512753805640\right) a{\left(n + 6 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(42151069746904 n^{5} + 8084275559102195 n^{4} + 286131378400632530 n^{3} + 4160668705504349845 n^{2} + 27584621698356560046 n + 69283710989690638320\right) a{\left(n + 10 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(105321626488373 n^{5} + 8379700935560035 n^{4} + 226027379163441565 n^{3} + 2795551294006591325 n^{2} + 16364874502918424742 n + 36857921492733813720\right) a{\left(n + 9 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(115077648643469 n^{5} + 28495025029972276 n^{4} + 2821099083762143879 n^{3} + 139585978246776293748 n^{2} + 3451690782090288874492 n + 34124837837691118427616\right) a{\left(n + 51 \right)}}{3808 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(625987084149583 n^{5} + 49416683683968155 n^{4} + 1427649669016888895 n^{3} + 19572653490458350705 n^{2} + 129600755298351587262 n + 334826647990454345400\right) a{\left(n + 11 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(1181965137196019 n^{5} + 73490161935689125 n^{4} + 1717266153800851015 n^{3} + 18781227442280706095 n^{2} + 94313729852835951546 n + 164953532971978759800\right) a{\left(n + 12 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1642527338207713 n^{5} + 183308518319935640 n^{4} + 6461805057486388835 n^{3} + 102379916226639884920 n^{2} + 762241308395440530252 n + 2177870140270111710960\right) a{\left(n + 13 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(5798234293573643 n^{5} + 449904551979214120 n^{4} + 13710070819377410925 n^{3} + 205879132649133325340 n^{2} + 1527766032175083355132 n + 4491689878711282130640\right) a{\left(n + 15 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(9766726316902699 n^{5} + 780970896636050957 n^{4} + 24669048245096580779 n^{3} + 384010242225839948851 n^{2} + 2939046723327473117682 n + 8825457214570345301304\right) a{\left(n + 19 \right)}}{5712 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(19506320672656154 n^{5} + 1295509887342498235 n^{4} + 34166228617473356020 n^{3} + 446810792049345193685 n^{2} + 2893506599369008165146 n + 7408996394128520895720\right) a{\left(n + 14 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(24157437766684273 n^{5} + 5905531802026906160 n^{4} + 577356627679268872675 n^{3} + 28217243841357158304400 n^{2} + 689395655335258488641332 n + 6735877168512407169674880\right) a{\left(n + 50 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(25166287465012813 n^{5} + 1895732160294196850 n^{4} + 55549915705182806095 n^{3} + 784794285685560354250 n^{2} + 5263951746017601532672 n + 12997082407355459069640\right) a{\left(n + 17 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(44233001996729426 n^{5} + 3609960082023220315 n^{4} + 115486944104238819140 n^{3} + 1796234230630787577395 n^{2} + 13403757930070295615864 n + 37429675500233154259020\right) a{\left(n + 20 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(63239791761031367 n^{5} + 4421457104075934025 n^{4} + 120913107004508963695 n^{3} + 1603668056694134220815 n^{2} + 10173449742158728682658 n + 24043728964202608687560\right) a{\left(n + 16 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(73348465414571133 n^{5} + 5873983316186517795 n^{4} + 186357154086327066885 n^{3} + 2922665817778207991845 n^{2} + 22604997779450136910142 n + 68750453288095272776640\right) a{\left(n + 18 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(92875799550841931 n^{5} + 12294864978844898290 n^{4} + 616871440942615640345 n^{3} + 14883731908670522895410 n^{2} + 174055717686088643973704 n + 792432298913016069160920\right) a{\left(n + 23 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(95798411319194161 n^{5} + 8460123475635077075 n^{4} + 292376592196499892055 n^{3} + 4902772116271359692375 n^{2} + 39342199825996099800774 n + 117690594242551202887740\right) a{\left(n + 22 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(117056043302411999 n^{5} + 5910216581931558310 n^{4} + 11132478101735250265 n^{3} - 4174189655177565382570 n^{2} - 84850194107929601802444 n - 509289128349932756875320\right) a{\left(n + 21 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(143391538030561528 n^{5} + 77256482753703734669 n^{4} + 10330195061184217621382 n^{3} + 590244796162053883024471 n^{2} + 15577385785752449393390718 n + 156743202067295451889956408\right) a{\left(n + 42 \right)}}{11424 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(145447369685589259 n^{5} + 18039199806025441280 n^{4} + 922148323289259143405 n^{3} + 24278971658023293216025 n^{2} + 328308751574540320327911 n + 1815760025610791182991910\right) a{\left(n + 26 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(146751103810381249 n^{5} + 35194095111317779010 n^{4} + 3375575757974202084215 n^{3} + 161854337122284890125870 n^{2} + 3879698752368006677347776 n + 37192687718816715849249840\right) a{\left(n + 49 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(419101140107127004 n^{5} + 50791505706013626635 n^{4} + 2429077716150675907530 n^{3} + 57216775970988049327205 n^{2} + 662251695245105441990386 n + 3002500500546931249471080\right) a{\left(n + 25 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(457497700760811543 n^{5} + 59708160300646452105 n^{4} + 3119037573164034241155 n^{3} + 81574021983602987332535 n^{2} + 1068988488254868026820902 n + 5620486413748454218532880\right) a{\left(n + 29 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(636900718048903262 n^{5} + 48380805839050866625 n^{4} + 651964361756601178720 n^{3} - 33253970631087679922065 n^{2} - 1104785195621814072319062 n - 9327192940934375279209920\right) a{\left(n + 28 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(700476453298091242 n^{5} + 130251806781261518063 n^{4} + 9659574567084048486100 n^{3} + 357205615831775182017091 n^{2} + 6587824056804349909828168 n + 48482378178902974682544348\right) a{\left(n + 37 \right)}}{1904 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(799110358748911633 n^{5} + 118962295829055951995 n^{4} + 6988725606355006265225 n^{3} + 202993622243368016797405 n^{2} + 2920179724571978594618862 n + 16666511532526755187128000\right) a{\left(n + 27 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(828132306056780422 n^{5} + 101116674595574681825 n^{4} + 4831957773440887106480 n^{3} + 113062148768404592275435 n^{2} + 1294909569761799318276798 n + 5797022355908741352536280\right) a{\left(n + 24 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1385018447342771233 n^{5} + 311112633299927101364 n^{4} + 27948477937438829249369 n^{3} + 1255122520608634366207924 n^{2} + 28177375245832980781880082 n + 252982450990770387940386864\right) a{\left(n + 46 \right)}}{5712 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1492905664686258644 n^{5} + 201500502920055851965 n^{4} + 10827561507949980553210 n^{3} + 289234228984403728264955 n^{2} + 3835459972348826721004746 n + 20159397096171161091728040\right) a{\left(n + 30 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1798780617799441669 n^{5} + 422322403344226393235 n^{4} + 39655417049331639232805 n^{3} + 1861497298639784878178005 n^{2} + 43684036873976206128730926 n + 409989312613868101519055760\right) a{\left(n + 48 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(1894942817768331298 n^{5} + 435090135452808023675 n^{4} + 39953007228247215839660 n^{3} + 1834069789403294753033965 n^{2} + 42089852325668933537636442 n + 386298571036483767822896160\right) a{\left(n + 47 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1976703052293726872 n^{5} + 345169184072168603365 n^{4} + 24189785244950100423880 n^{3} + 849340684805344027316405 n^{2} + 14925183963504568807656918 n + 104923119378559478299852140\right) a{\left(n + 33 \right)}}{28560 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(2018405104038600077 n^{5} + 273631598689776718135 n^{4} + 14878577093397634669585 n^{3} + 406531071021707759801525 n^{2} + 5597413234443645356424558 n + 31171905756142306033358880\right) a{\left(n + 31 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(3025048152504877649 n^{5} + 457522788808283372830 n^{4} + 27886766526384274335055 n^{3} + 856642477594086945708230 n^{2} + 13265967841716696904846596 n + 82859085908346531414811140\right) a{\left(n + 32 \right)}}{28560 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(3151391850559835963 n^{5} + 595103950320060532540 n^{4} + 44711322740308889934805 n^{3} + 1671927257535257716867930 n^{2} + 31135839664108476411672502 n + 231130440391022009628457740\right) a{\left(n + 36 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(5645307042197447978 n^{5} + 1216960941578987755480 n^{4} + 104869794590526891116185 n^{3} + 4515743860964415650454545 n^{2} + 97167992555946819332912862 n + 835856550451893143507350950\right) a{\left(n + 43 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(6121351114203904037 n^{5} + 1324860573732304348198 n^{4} + 114671768150252347776547 n^{3} + 4961536168335050085422342 n^{2} + 107312189014333472882473212 n + 928207685454030156725261640\right) a{\left(n + 44 \right)}}{11424 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(6574002295115052656 n^{5} + 1273365966617326864645 n^{4} + 96958208914594028563570 n^{3} + 3641665004036798136664235 n^{2} + 67650174923322848656014414 n + 498234480988548694813592400\right) a{\left(n + 34 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(7344935948635588041 n^{5} + 1357069956476567414460 n^{4} + 99314194648930556470835 n^{3} + 3590387776407173586316620 n^{2} + 63917435878706639167270924 n + 446207638140435549403181760\right) a{\left(n + 41 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11275810966190104019 n^{5} + 2190856846195136540620 n^{4} + 167884587833785317625465 n^{3} + 6360939061679931214313660 n^{2} + 119416667319079082180567076 n + 890056764377605586765489040\right) a{\left(n + 35 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(24679889108363750194 n^{5} + 5432771223146272833905 n^{4} + 478270882837908026885780 n^{3} + 21047958131362307484036055 n^{2} + 463050869305847640685364706 n + 4073984264783362269255421680\right) a{\left(n + 45 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(25055180656927910492 n^{5} + 4555520019760066368745 n^{4} + 330037428207408065301910 n^{3} + 11905252459187871674672375 n^{2} + 213740180728221557205206838 n + 1527128575179148558794329280\right) a{\left(n + 38 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(30161915058115285133 n^{5} + 5459750116319275759915 n^{4} + 392785943648326592508685 n^{3} + 14023103460820180614220865 n^{2} + 248097365726798410941112002 n + 1736851058316966101200870680\right) a{\left(n + 39 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(33028307806250899492 n^{5} + 6078170135984892569195 n^{4} + 444412061686255894217210 n^{3} + 16117516181691819382525945 n^{2} + 289471790474437443720578718 n + 2055200791696009176875435280\right) a{\left(n + 40 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)}, \quad n \geq 57\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 517\)
\(\displaystyle a(7) = 2580\)
\(\displaystyle a(8) = 13277\)
\(\displaystyle a(9) = 69956\)
\(\displaystyle a(10) = 375523\)
\(\displaystyle a(11) = 2046393\)
\(\displaystyle a(12) = 11291232\)
\(\displaystyle a(13) = 62954575\)
\(\displaystyle a(14) = 354139163\)
\(\displaystyle a(15) = 2007464860\)
\(\displaystyle a(16) = 11455640891\)
\(\displaystyle a(17) = 65756198379\)
\(\displaystyle a(18) = 379409903992\)
\(\displaystyle a(19) = 2199334938209\)
\(\displaystyle a(20) = 12802019416555\)
\(\displaystyle a(21) = 74798669115432\)
\(\displaystyle a(22) = 438516581229747\)
\(\displaystyle a(23) = 2578835415720416\)
\(\displaystyle a(24) = 15208702319342870\)
\(\displaystyle a(25) = 89927210626144019\)
\(\displaystyle a(26) = 533005690952120498\)
\(\displaystyle a(27) = 3166180739828123982\)
\(\displaystyle a(28) = 18846584060274483157\)
\(\displaystyle a(29) = 112398367697187844181\)
\(\displaystyle a(30) = 671523911133583255448\)
\(\displaystyle a(31) = 4018706774817784236602\)
\(\displaystyle a(32) = 24087326202516995486315\)
\(\displaystyle a(33) = 144586126453184080411950\)
\(\displaystyle a(34) = 869085401365072758359137\)
\(\displaystyle a(35) = 5230719600455432818148141\)
\(\displaystyle a(36) = 31520409831325553252519341\)
\(\displaystyle a(37) = 190162372803590296056952175\)
\(\displaystyle a(38) = 1148504895789457613160868957\)
\(\displaystyle a(39) = 6943715945734894247920438085\)
\(\displaystyle a(40) = 42022225778307662507192051567\)
\(\displaystyle a(41) = 254549961137867869415255024247\)
\(\displaystyle a(42) = 1543313984587182195685210289899\)
\(\displaystyle a(43) = 9364931397955286156486818280632\)
\(\displaystyle a(44) = 56873112577757474259849429633394\)
\(\displaystyle a(45) = 345657242683256160441523582087493\)
\(\displaystyle a(46) = 2102353851295376800979994736917069\)
\(\displaystyle a(47) = 12795982379336775491182499776854016\)
\(\displaystyle a(48) = 77935650654475916813024286490921993\)
\(\displaystyle a(49) = 474986493951181041883645820199913714\)
\(\displaystyle a(50) = 2896659886230297525360326238757884004\)
\(\displaystyle a(51) = 17675599896983326737783387011002382768\)
\(\displaystyle a(52) = 107919797467564943383654218940807259215\)
\(\displaystyle a(53) = 659278404034369402375256144570834899233\)
\(\displaystyle a(54) = 4029659479143810004063337689269488381852\)
\(\displaystyle a(55) = 24642860213918857626042963054884352530838\)
\(\displaystyle a(56) = 150774918311369904033493038993550563200134\)
\(\displaystyle a{\left(n + 57 \right)} = \frac{2193632 n \left(n - 1\right) \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{255 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{56 n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(308311 n + 1528964\right) a{\left(n + 1 \right)}}{51 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{4 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(41264566 n^{2} + 492378809 n + 1392163335\right) a{\left(n + 2 \right)}}{255 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(n + 2\right) \left(n + 3\right) \left(9304000366 n^{3} + 166265700750 n^{2} + 933626654579 n + 1670744516340\right) a{\left(n + 3 \right)}}{1785 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(n + 3\right) \left(163759458917 n^{4} + 3999140808064 n^{3} + 35759941038013 n^{2} + 138376022277866 n + 195392634704280\right) a{\left(n + 4 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(36429503 n^{5} + 10123667410 n^{4} + 1125306256805 n^{3} + 62540562021650 n^{2} + 1737844576161512 n + 19315595597188320\right) a{\left(n + 56 \right)}}{170 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(3897317477 n^{5} + 661969065250 n^{4} + 29917714105535 n^{3} - 598336751858150 n^{2} - 75090905798490312 n - 1438444165524413520\right) a{\left(n + 54 \right)}}{4760 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(55579306351 n^{5} + 15172113565880 n^{4} + 1656666951679805 n^{3} + 90446257184215840 n^{2} + 2468950325947482204 n + 26958178791399337200\right) a{\left(n + 55 \right)}}{7140 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(275620558993 n^{5} + 8683164741110 n^{4} + 109936565938495 n^{3} + 689209032103810 n^{2} + 2121777905252192 n + 2555051413977120\right) a{\left(n + 5 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(10094380043275 n^{5} + 2627468565737690 n^{4} + 273561113348769565 n^{3} + 14240958900222810806 n^{2} + 370673232619048084224 n + 3859219047263196299616\right) a{\left(n + 52 \right)}}{1904 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11048636087637 n^{5} + 2883892018243620 n^{4} + 301092030515000555 n^{3} + 15717373612054815120 n^{2} + 410224565435420680588 n + 4282672360040962697520\right) a{\left(n + 53 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11905948444723 n^{5} + 496318654712570 n^{4} + 8377694457189115 n^{3} + 71152983544547710 n^{2} + 302658214512471562 n + 514109679835712340\right) a{\left(n + 8 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(13672278486428 n^{5} + 686160433491055 n^{4} + 12993775697642550 n^{3} + 117764901650282545 n^{2} + 515490304286908022 n + 877120393864696320\right) a{\left(n + 7 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(20510953891186 n^{5} + 796930701558185 n^{4} + 12208339786699580 n^{3} + 91760290676249515 n^{2} + 337755239332506774 n + 486827512753805640\right) a{\left(n + 6 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(42151069746904 n^{5} + 8084275559102195 n^{4} + 286131378400632530 n^{3} + 4160668705504349845 n^{2} + 27584621698356560046 n + 69283710989690638320\right) a{\left(n + 10 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(105321626488373 n^{5} + 8379700935560035 n^{4} + 226027379163441565 n^{3} + 2795551294006591325 n^{2} + 16364874502918424742 n + 36857921492733813720\right) a{\left(n + 9 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(115077648643469 n^{5} + 28495025029972276 n^{4} + 2821099083762143879 n^{3} + 139585978246776293748 n^{2} + 3451690782090288874492 n + 34124837837691118427616\right) a{\left(n + 51 \right)}}{3808 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(625987084149583 n^{5} + 49416683683968155 n^{4} + 1427649669016888895 n^{3} + 19572653490458350705 n^{2} + 129600755298351587262 n + 334826647990454345400\right) a{\left(n + 11 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(1181965137196019 n^{5} + 73490161935689125 n^{4} + 1717266153800851015 n^{3} + 18781227442280706095 n^{2} + 94313729852835951546 n + 164953532971978759800\right) a{\left(n + 12 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1642527338207713 n^{5} + 183308518319935640 n^{4} + 6461805057486388835 n^{3} + 102379916226639884920 n^{2} + 762241308395440530252 n + 2177870140270111710960\right) a{\left(n + 13 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(5798234293573643 n^{5} + 449904551979214120 n^{4} + 13710070819377410925 n^{3} + 205879132649133325340 n^{2} + 1527766032175083355132 n + 4491689878711282130640\right) a{\left(n + 15 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(9766726316902699 n^{5} + 780970896636050957 n^{4} + 24669048245096580779 n^{3} + 384010242225839948851 n^{2} + 2939046723327473117682 n + 8825457214570345301304\right) a{\left(n + 19 \right)}}{5712 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(19506320672656154 n^{5} + 1295509887342498235 n^{4} + 34166228617473356020 n^{3} + 446810792049345193685 n^{2} + 2893506599369008165146 n + 7408996394128520895720\right) a{\left(n + 14 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(24157437766684273 n^{5} + 5905531802026906160 n^{4} + 577356627679268872675 n^{3} + 28217243841357158304400 n^{2} + 689395655335258488641332 n + 6735877168512407169674880\right) a{\left(n + 50 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(25166287465012813 n^{5} + 1895732160294196850 n^{4} + 55549915705182806095 n^{3} + 784794285685560354250 n^{2} + 5263951746017601532672 n + 12997082407355459069640\right) a{\left(n + 17 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(44233001996729426 n^{5} + 3609960082023220315 n^{4} + 115486944104238819140 n^{3} + 1796234230630787577395 n^{2} + 13403757930070295615864 n + 37429675500233154259020\right) a{\left(n + 20 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(63239791761031367 n^{5} + 4421457104075934025 n^{4} + 120913107004508963695 n^{3} + 1603668056694134220815 n^{2} + 10173449742158728682658 n + 24043728964202608687560\right) a{\left(n + 16 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(73348465414571133 n^{5} + 5873983316186517795 n^{4} + 186357154086327066885 n^{3} + 2922665817778207991845 n^{2} + 22604997779450136910142 n + 68750453288095272776640\right) a{\left(n + 18 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(92875799550841931 n^{5} + 12294864978844898290 n^{4} + 616871440942615640345 n^{3} + 14883731908670522895410 n^{2} + 174055717686088643973704 n + 792432298913016069160920\right) a{\left(n + 23 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(95798411319194161 n^{5} + 8460123475635077075 n^{4} + 292376592196499892055 n^{3} + 4902772116271359692375 n^{2} + 39342199825996099800774 n + 117690594242551202887740\right) a{\left(n + 22 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(117056043302411999 n^{5} + 5910216581931558310 n^{4} + 11132478101735250265 n^{3} - 4174189655177565382570 n^{2} - 84850194107929601802444 n - 509289128349932756875320\right) a{\left(n + 21 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(143391538030561528 n^{5} + 77256482753703734669 n^{4} + 10330195061184217621382 n^{3} + 590244796162053883024471 n^{2} + 15577385785752449393390718 n + 156743202067295451889956408\right) a{\left(n + 42 \right)}}{11424 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(145447369685589259 n^{5} + 18039199806025441280 n^{4} + 922148323289259143405 n^{3} + 24278971658023293216025 n^{2} + 328308751574540320327911 n + 1815760025610791182991910\right) a{\left(n + 26 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(146751103810381249 n^{5} + 35194095111317779010 n^{4} + 3375575757974202084215 n^{3} + 161854337122284890125870 n^{2} + 3879698752368006677347776 n + 37192687718816715849249840\right) a{\left(n + 49 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(419101140107127004 n^{5} + 50791505706013626635 n^{4} + 2429077716150675907530 n^{3} + 57216775970988049327205 n^{2} + 662251695245105441990386 n + 3002500500546931249471080\right) a{\left(n + 25 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{3 \left(457497700760811543 n^{5} + 59708160300646452105 n^{4} + 3119037573164034241155 n^{3} + 81574021983602987332535 n^{2} + 1068988488254868026820902 n + 5620486413748454218532880\right) a{\left(n + 29 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(636900718048903262 n^{5} + 48380805839050866625 n^{4} + 651964361756601178720 n^{3} - 33253970631087679922065 n^{2} - 1104785195621814072319062 n - 9327192940934375279209920\right) a{\left(n + 28 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(700476453298091242 n^{5} + 130251806781261518063 n^{4} + 9659574567084048486100 n^{3} + 357205615831775182017091 n^{2} + 6587824056804349909828168 n + 48482378178902974682544348\right) a{\left(n + 37 \right)}}{1904 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(799110358748911633 n^{5} + 118962295829055951995 n^{4} + 6988725606355006265225 n^{3} + 202993622243368016797405 n^{2} + 2920179724571978594618862 n + 16666511532526755187128000\right) a{\left(n + 27 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(828132306056780422 n^{5} + 101116674595574681825 n^{4} + 4831957773440887106480 n^{3} + 113062148768404592275435 n^{2} + 1294909569761799318276798 n + 5797022355908741352536280\right) a{\left(n + 24 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1385018447342771233 n^{5} + 311112633299927101364 n^{4} + 27948477937438829249369 n^{3} + 1255122520608634366207924 n^{2} + 28177375245832980781880082 n + 252982450990770387940386864\right) a{\left(n + 46 \right)}}{5712 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1492905664686258644 n^{5} + 201500502920055851965 n^{4} + 10827561507949980553210 n^{3} + 289234228984403728264955 n^{2} + 3835459972348826721004746 n + 20159397096171161091728040\right) a{\left(n + 30 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1798780617799441669 n^{5} + 422322403344226393235 n^{4} + 39655417049331639232805 n^{3} + 1861497298639784878178005 n^{2} + 43684036873976206128730926 n + 409989312613868101519055760\right) a{\left(n + 48 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(1894942817768331298 n^{5} + 435090135452808023675 n^{4} + 39953007228247215839660 n^{3} + 1834069789403294753033965 n^{2} + 42089852325668933537636442 n + 386298571036483767822896160\right) a{\left(n + 47 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(1976703052293726872 n^{5} + 345169184072168603365 n^{4} + 24189785244950100423880 n^{3} + 849340684805344027316405 n^{2} + 14925183963504568807656918 n + 104923119378559478299852140\right) a{\left(n + 33 \right)}}{28560 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(2018405104038600077 n^{5} + 273631598689776718135 n^{4} + 14878577093397634669585 n^{3} + 406531071021707759801525 n^{2} + 5597413234443645356424558 n + 31171905756142306033358880\right) a{\left(n + 31 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(3025048152504877649 n^{5} + 457522788808283372830 n^{4} + 27886766526384274335055 n^{3} + 856642477594086945708230 n^{2} + 13265967841716696904846596 n + 82859085908346531414811140\right) a{\left(n + 32 \right)}}{28560 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(3151391850559835963 n^{5} + 595103950320060532540 n^{4} + 44711322740308889934805 n^{3} + 1671927257535257716867930 n^{2} + 31135839664108476411672502 n + 231130440391022009628457740\right) a{\left(n + 36 \right)}}{9520 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(5645307042197447978 n^{5} + 1216960941578987755480 n^{4} + 104869794590526891116185 n^{3} + 4515743860964415650454545 n^{2} + 97167992555946819332912862 n + 835856550451893143507350950\right) a{\left(n + 43 \right)}}{14280 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(6121351114203904037 n^{5} + 1324860573732304348198 n^{4} + 114671768150252347776547 n^{3} + 4961536168335050085422342 n^{2} + 107312189014333472882473212 n + 928207685454030156725261640\right) a{\left(n + 44 \right)}}{11424 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(6574002295115052656 n^{5} + 1273365966617326864645 n^{4} + 96958208914594028563570 n^{3} + 3641665004036798136664235 n^{2} + 67650174923322848656014414 n + 498234480988548694813592400\right) a{\left(n + 34 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(7344935948635588041 n^{5} + 1357069956476567414460 n^{4} + 99314194648930556470835 n^{3} + 3590387776407173586316620 n^{2} + 63917435878706639167270924 n + 446207638140435549403181760\right) a{\left(n + 41 \right)}}{19040 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(11275810966190104019 n^{5} + 2190856846195136540620 n^{4} + 167884587833785317625465 n^{3} + 6360939061679931214313660 n^{2} + 119416667319079082180567076 n + 890056764377605586765489040\right) a{\left(n + 35 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(24679889108363750194 n^{5} + 5432771223146272833905 n^{4} + 478270882837908026885780 n^{3} + 21047958131362307484036055 n^{2} + 463050869305847640685364706 n + 4073984264783362269255421680\right) a{\left(n + 45 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(25055180656927910492 n^{5} + 4555520019760066368745 n^{4} + 330037428207408065301910 n^{3} + 11905252459187871674672375 n^{2} + 213740180728221557205206838 n + 1527128575179148558794329280\right) a{\left(n + 38 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} - \frac{\left(30161915058115285133 n^{5} + 5459750116319275759915 n^{4} + 392785943648326592508685 n^{3} + 14023103460820180614220865 n^{2} + 248097365726798410941112002 n + 1736851058316966101200870680\right) a{\left(n + 39 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)} + \frac{\left(33028307806250899492 n^{5} + 6078170135984892569195 n^{4} + 444412061686255894217210 n^{3} + 16117516181691819382525945 n^{2} + 289471790474437443720578718 n + 2055200791696009176875435280\right) a{\left(n + 40 \right)}}{57120 \left(n + 57\right) \left(5 n + 282\right) \left(5 n + 283\right) \left(5 n + 284\right) \left(5 n + 286\right)}, \quad n \geq 57\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 50 rules.
Finding the specification took 1875 seconds.
Copy 50 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_{16}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{48}\! \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_{16}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 49 rules.
Finding the specification took 863 seconds.
Copy 49 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_{16}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{47}\! \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_{16}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)