Av(13542, 14523, 14532, 15423, 15432, 24513, 25413)
Counting Sequence
1, 1, 2, 6, 24, 113, 581, 3139, 17500, 99772, 578776, 3405089, 20269718, 121869038, 739015133, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x -1\right) F \left(x
\right)^{5}-x^{2} \left(x -1\right) \left(2 x -3\right) F \left(x
\right)^{4}+\left(-3 x^{3}+5 x^{2}+x +1\right) F \left(x
\right)^{3}+\left(-x^{2}-2 x -5\right) F \left(x
\right)^{2}+8 F \! \left(x \right)-4 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 581\)
\(\displaystyle a(7) = 3139\)
\(\displaystyle a(8) = 17500\)
\(\displaystyle a(9) = 99772\)
\(\displaystyle a(10) = 578776\)
\(\displaystyle a(11) = 3405089\)
\(\displaystyle a(12) = 20269718\)
\(\displaystyle a(13) = 121869038\)
\(\displaystyle a(14) = 739015133\)
\(\displaystyle a(15) = 4514754917\)
\(\displaystyle a(16) = 27760850118\)
\(\displaystyle a(17) = 171678490387\)
\(\displaystyle a(18) = 1067100692285\)
\(\displaystyle a(19) = 6662955903479\)
\(\displaystyle a(20) = 41773543226571\)
\(\displaystyle a(21) = 262868106119301\)
\(\displaystyle a(22) = 1659699442472225\)
\(\displaystyle a(23) = 10511097379593102\)
\(\displaystyle a(24) = 66754711794724532\)
\(\displaystyle a(25) = 425042894136874547\)
\(\displaystyle a(26) = 2712774605779759537\)
\(\displaystyle a(27) = 17351924803873950301\)
\(\displaystyle a(28) = 111215625138092765752\)
\(\displaystyle a(29) = 714178134717729765251\)
\(\displaystyle a(30) = 4594248529257237170046\)
\(\displaystyle a(31) = 29603265548417261359584\)
\(\displaystyle a(32) = 191045446833411670337024\)
\(\displaystyle a(33) = 1234708906787529804034607\)
\(\displaystyle a(34) = 7990718489586348724436119\)
\(\displaystyle a(35) = 51780497680111387335752242\)
\(\displaystyle a(36) = 335949812589365341220112511\)
\(\displaystyle a(37) = 2182134901189588893803852599\)
\(\displaystyle a(38) = 14189310870605689682019965570\)
\(\displaystyle a(39) = 92361098172145492590672590974\)
\(\displaystyle a(40) = 601786382315511826292004432973\)
\(\displaystyle a(41) = 3924643205381700198106970393195\)
\(\displaystyle a(42) = 25617877283422343005358956650902\)
\(\displaystyle a(43) = 167360583381100994034720642578192\)
\(\displaystyle a(44) = 1094242264018218441687029729916607\)
\(\displaystyle a(45) = 7159923231160759605436958332402913\)
\(\displaystyle a(46) = 46883839122967264822577159404458662\)
\(\displaystyle a(47) = 307216243188291654497696144589684371\)
\(\displaystyle a(48) = 2014459476187210445480157726964631976\)
\(\displaystyle a{\left(n + 49 \right)} = - \frac{3767606827189 n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{5 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{2021 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(28238316830 n + 107365190291\right) a{\left(n + 1 \right)}}{5 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{3 \left(n + 2\right) \left(n + 3\right) \left(514277482176014 n^{2} + 4405429126785254 n + 9396926337982697\right) a{\left(n + 2 \right)}}{20 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(n + 3\right) \left(2937850015385398 n^{3} + 41740968971361747 n^{2} + 196587669037905707 n + 306509744999905245\right) a{\left(n + 3 \right)}}{10 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(851 n^{2} + 81903 n + 1969057\right) a{\left(n + 48 \right)}}{10 \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(57338 n^{3} + 8060845 n^{2} + 377574589 n + 5892594692\right) a{\left(n + 47 \right)}}{40 \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(765811 n^{4} + 139517384 n^{3} + 9523243640 n^{2} + 288640063147 n + 3277437749970\right) a{\left(n + 46 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(3239891 n^{4} + 619149338 n^{3} + 44302284700 n^{2} + 1406874587317 n + 16731646684614\right) a{\left(n + 45 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(410241945 n^{4} + 73443528770 n^{3} + 4930700188062 n^{2} + 147126946186693 n + 1646337844515876\right) a{\left(n + 44 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(3394459894 n^{4} + 493921276668 n^{3} + 26114058309077 n^{2} + 585579572412111 n + 4560244271972298\right) a{\left(n + 42 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(6136991490 n^{4} + 1057759121669 n^{3} + 68349310008075 n^{2} + 1962375602418502 n + 21122381501350296\right) a{\left(n + 43 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(237170405486 n^{4} + 40442025214215 n^{3} + 2584924850489563 n^{2} + 73400604469397862 n + 781284456669850368\right) a{\left(n + 41 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(1004333031115 n^{4} + 163898321148717 n^{3} + 10029159749444384 n^{2} + 272732585882700660 n + 2781049349231249928\right) a{\left(n + 40 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(1397473958326 n^{4} + 216563773470975 n^{3} + 12571932477254642 n^{2} + 324010212807851922 n + 3127864620193062894\right) a{\left(n + 39 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(11161090486313 n^{4} + 1815303106738325 n^{3} + 110542506258176710 n^{2} + 2987274516076916536 n + 30229987343378104218\right) a{\left(n + 38 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(90749746266249 n^{4} + 13075661796062555 n^{3} + 708359480637454974 n^{2} + 17101331772601241991 n + 155247470590497361588\right) a{\left(n + 35 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(265469800314274 n^{4} + 40789516164065719 n^{3} + 2350314767739207341 n^{2} + 60190921234148936774 n + 578062063886063472768\right) a{\left(n + 37 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(496464566823943 n^{4} + 73838093138246014 n^{3} + 4119346286012481623 n^{2} + 102168753070017360608 n + 950525042230363967586\right) a{\left(n + 36 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(714533808369091 n^{4} - 29170025834485518 n^{3} - 672308323503434621 n^{2} - 4262107798247729400 n - 8596016517272382176\right) a{\left(n + 5 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(11285111271059802 n^{4} + 1587693168947198365 n^{3} + 83734400068220877903 n^{2} + 1961997309167643272510 n + 17232883764997469888052\right) a{\left(n + 34 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(23592405948071158 n^{4} + 1838554672523878175 n^{3} + 29879701426565782313 n^{2} - 644276118975941360600 n - 15706408913344418725272\right) a{\left(n + 31 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(24027875084673232 n^{4} + 486731333139426320 n^{3} + 3677414278520319857 n^{2} + 12271910992242903751 n + 15248540530093316922\right) a{\left(n + 4 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(27383951364279680 n^{4} + 3756858132200832032 n^{3} + 193261896081863137864 n^{2} + 4418187540507889524733 n + 37873004630472380615442\right) a{\left(n + 33 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(109905585467737568 n^{4} + 14852840585693894845 n^{3} + 752619095443594533943 n^{2} + 16947387913246928559242 n + 143088766327097185851096\right) a{\left(n + 32 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(161426699054292668 n^{4} + 5252482297962464353 n^{3} + 62617357824753755230 n^{2} + 325330818680209958633 n + 623086556997783459072\right) a{\left(n + 6 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(474408101665695936 n^{4} + 56721906799189000642 n^{3} + 2540210323419533589477 n^{2} + 50497385327601946223315 n + 375952878720834816437982\right) a{\left(n + 29 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(742438770388733784 n^{4} + 89698100526323982521 n^{3} + 4055653962786953045439 n^{2} + 81319510782751927686076 n + 609959168103285776928468\right) a{\left(n + 30 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(1095182150231720140 n^{4} + 138658618965785370519 n^{3} + 6520871903734645989581 n^{2} + 135201396315248102130714 n + 1043929238413855618597404\right) a{\left(n + 28 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(2434236057501410494 n^{4} + 86821636393050800484 n^{3} + 1146455464407518473355 n^{2} + 6649674628467200823963 n + 14306486978573823934464\right) a{\left(n + 7 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(2661641226876060215 n^{4} + 105516395853346770897 n^{3} + 1553909654577380284381 n^{2} + 10079684462182074943584 n + 24309239015213590584330\right) a{\left(n + 8 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(3267432211933192831 n^{4} + 454271154724284504226 n^{3} + 20769133745246058246587 n^{2} + 393417653024990181936932 n + 2674503247666513338315138\right) a{\left(n + 22 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(6614535911630723275 n^{4} + 171385913342386364616 n^{3} + 406312218740865612701 n^{2} - 18404424759873275742414 n - 120363671111476080756624\right) a{\left(n + 11 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(6850572697097133572 n^{4} + 724124605053318632687 n^{3} + 28603108980924412402003 n^{2} + 500183768713666587529378 n + 3265546328358626478845016\right) a{\left(n + 27 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(8377681239583737686 n^{4} + 651207869789535996117 n^{3} + 18768470992765264520783 n^{2} + 237161978830493576296698 n + 1105167422620684615662384\right) a{\left(n + 21 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(11031482531251448377 n^{4} + 842369726409146808156 n^{3} + 24300757366847579320775 n^{2} + 314359672417914189719357 n + 1540803401192894500994300\right) a{\left(n + 19 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(11849150073753485332 n^{4} + 615321778587984032929 n^{3} + 11738850311502705569777 n^{2} + 97836660472256368847198 n + 301313855560040744395428\right) a{\left(n + 10 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(12766544158333342024 n^{4} + 1337346070708452471074 n^{3} + 52454696318682520561571 n^{2} + 912913282502345706496477 n + 5947502795228955649329582\right) a{\left(n + 26 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(15282750157919709658 n^{4} + 676333184657934871960 n^{3} + 11126371380199943137493 n^{2} + 80667192327398566316519 n + 217542936758213717129538\right) a{\left(n + 9 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(21171084178172935298 n^{4} + 2124283604158655523459 n^{3} + 79561476649751392136917 n^{2} + 1318824022674310040831946 n + 8166423963766963017349524\right) a{\left(n + 23 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(21541903879397745871 n^{4} + 1745185370080318340869 n^{3} + 53239846233118006589828 n^{2} + 725475520210016059567118 n + 3728900035826397129701310\right) a{\left(n + 20 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(29466424275665033319 n^{4} + 2951784723011174636842 n^{3} + 110714475255371122912767 n^{2} + 1842745061723509897002782 n + 11483405386731997642324344\right) a{\left(n + 24 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(31832105202637336254 n^{4} + 2007106207682007427653 n^{3} + 47141568470015328133035 n^{2} + 488955292286476589806141 n + 1891097593940168132554214\right) a{\left(n + 17 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(48700903523066779442 n^{4} + 4981077020721326249687 n^{3} + 190809709526821970525053 n^{2} + 3244315214119336658097664 n + 20657104025398178364162108\right) a{\left(n + 25 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(82354825250731340918 n^{4} + 3724559200543369105071 n^{3} + 61864092715493979062977 n^{2} + 444099017570051223996048 n + 1148981115502212550117308\right) a{\left(n + 12 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(83806689305070961418 n^{4} + 4146849813159875182881 n^{3} + 76018177169995886009785 n^{2} + 610065169431970924628556 n + 1800766835236262656263396\right) a{\left(n + 13 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(117369035455142427782 n^{4} + 8090580405278867449051 n^{3} + 209636772756552719247901 n^{2} + 2425488190666036597495574 n + 10602819313907175957584772\right) a{\left(n + 18 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(240535095690493951244 n^{4} + 12617951173375116933505 n^{3} + 245122423418992007761243 n^{2} + 2083779039284955893950628 n + 6512469690773607512113932\right) a{\left(n + 14 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(253858678708561257756 n^{4} + 14907049415478806209699 n^{3} + 324196655836613006826573 n^{2} + 3087766005343445816352110 n + 10834551047862484950834216\right) a{\left(n + 16 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(273720309364049278174 n^{4} + 15148953482930377420051 n^{3} + 310089612866803977082805 n^{2} + 2773299349124586372682304 n + 9101093245329040426087032\right) a{\left(n + 15 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)}, \quad n \geq 49\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 581\)
\(\displaystyle a(7) = 3139\)
\(\displaystyle a(8) = 17500\)
\(\displaystyle a(9) = 99772\)
\(\displaystyle a(10) = 578776\)
\(\displaystyle a(11) = 3405089\)
\(\displaystyle a(12) = 20269718\)
\(\displaystyle a(13) = 121869038\)
\(\displaystyle a(14) = 739015133\)
\(\displaystyle a(15) = 4514754917\)
\(\displaystyle a(16) = 27760850118\)
\(\displaystyle a(17) = 171678490387\)
\(\displaystyle a(18) = 1067100692285\)
\(\displaystyle a(19) = 6662955903479\)
\(\displaystyle a(20) = 41773543226571\)
\(\displaystyle a(21) = 262868106119301\)
\(\displaystyle a(22) = 1659699442472225\)
\(\displaystyle a(23) = 10511097379593102\)
\(\displaystyle a(24) = 66754711794724532\)
\(\displaystyle a(25) = 425042894136874547\)
\(\displaystyle a(26) = 2712774605779759537\)
\(\displaystyle a(27) = 17351924803873950301\)
\(\displaystyle a(28) = 111215625138092765752\)
\(\displaystyle a(29) = 714178134717729765251\)
\(\displaystyle a(30) = 4594248529257237170046\)
\(\displaystyle a(31) = 29603265548417261359584\)
\(\displaystyle a(32) = 191045446833411670337024\)
\(\displaystyle a(33) = 1234708906787529804034607\)
\(\displaystyle a(34) = 7990718489586348724436119\)
\(\displaystyle a(35) = 51780497680111387335752242\)
\(\displaystyle a(36) = 335949812589365341220112511\)
\(\displaystyle a(37) = 2182134901189588893803852599\)
\(\displaystyle a(38) = 14189310870605689682019965570\)
\(\displaystyle a(39) = 92361098172145492590672590974\)
\(\displaystyle a(40) = 601786382315511826292004432973\)
\(\displaystyle a(41) = 3924643205381700198106970393195\)
\(\displaystyle a(42) = 25617877283422343005358956650902\)
\(\displaystyle a(43) = 167360583381100994034720642578192\)
\(\displaystyle a(44) = 1094242264018218441687029729916607\)
\(\displaystyle a(45) = 7159923231160759605436958332402913\)
\(\displaystyle a(46) = 46883839122967264822577159404458662\)
\(\displaystyle a(47) = 307216243188291654497696144589684371\)
\(\displaystyle a(48) = 2014459476187210445480157726964631976\)
\(\displaystyle a{\left(n + 49 \right)} = - \frac{3767606827189 n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{5 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{2021 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(28238316830 n + 107365190291\right) a{\left(n + 1 \right)}}{5 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{3 \left(n + 2\right) \left(n + 3\right) \left(514277482176014 n^{2} + 4405429126785254 n + 9396926337982697\right) a{\left(n + 2 \right)}}{20 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(n + 3\right) \left(2937850015385398 n^{3} + 41740968971361747 n^{2} + 196587669037905707 n + 306509744999905245\right) a{\left(n + 3 \right)}}{10 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(851 n^{2} + 81903 n + 1969057\right) a{\left(n + 48 \right)}}{10 \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(57338 n^{3} + 8060845 n^{2} + 377574589 n + 5892594692\right) a{\left(n + 47 \right)}}{40 \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(765811 n^{4} + 139517384 n^{3} + 9523243640 n^{2} + 288640063147 n + 3277437749970\right) a{\left(n + 46 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(3239891 n^{4} + 619149338 n^{3} + 44302284700 n^{2} + 1406874587317 n + 16731646684614\right) a{\left(n + 45 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(410241945 n^{4} + 73443528770 n^{3} + 4930700188062 n^{2} + 147126946186693 n + 1646337844515876\right) a{\left(n + 44 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(3394459894 n^{4} + 493921276668 n^{3} + 26114058309077 n^{2} + 585579572412111 n + 4560244271972298\right) a{\left(n + 42 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(6136991490 n^{4} + 1057759121669 n^{3} + 68349310008075 n^{2} + 1962375602418502 n + 21122381501350296\right) a{\left(n + 43 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(237170405486 n^{4} + 40442025214215 n^{3} + 2584924850489563 n^{2} + 73400604469397862 n + 781284456669850368\right) a{\left(n + 41 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(1004333031115 n^{4} + 163898321148717 n^{3} + 10029159749444384 n^{2} + 272732585882700660 n + 2781049349231249928\right) a{\left(n + 40 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(1397473958326 n^{4} + 216563773470975 n^{3} + 12571932477254642 n^{2} + 324010212807851922 n + 3127864620193062894\right) a{\left(n + 39 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(11161090486313 n^{4} + 1815303106738325 n^{3} + 110542506258176710 n^{2} + 2987274516076916536 n + 30229987343378104218\right) a{\left(n + 38 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(90749746266249 n^{4} + 13075661796062555 n^{3} + 708359480637454974 n^{2} + 17101331772601241991 n + 155247470590497361588\right) a{\left(n + 35 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(265469800314274 n^{4} + 40789516164065719 n^{3} + 2350314767739207341 n^{2} + 60190921234148936774 n + 578062063886063472768\right) a{\left(n + 37 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(496464566823943 n^{4} + 73838093138246014 n^{3} + 4119346286012481623 n^{2} + 102168753070017360608 n + 950525042230363967586\right) a{\left(n + 36 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(714533808369091 n^{4} - 29170025834485518 n^{3} - 672308323503434621 n^{2} - 4262107798247729400 n - 8596016517272382176\right) a{\left(n + 5 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(11285111271059802 n^{4} + 1587693168947198365 n^{3} + 83734400068220877903 n^{2} + 1961997309167643272510 n + 17232883764997469888052\right) a{\left(n + 34 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(23592405948071158 n^{4} + 1838554672523878175 n^{3} + 29879701426565782313 n^{2} - 644276118975941360600 n - 15706408913344418725272\right) a{\left(n + 31 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(24027875084673232 n^{4} + 486731333139426320 n^{3} + 3677414278520319857 n^{2} + 12271910992242903751 n + 15248540530093316922\right) a{\left(n + 4 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(27383951364279680 n^{4} + 3756858132200832032 n^{3} + 193261896081863137864 n^{2} + 4418187540507889524733 n + 37873004630472380615442\right) a{\left(n + 33 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(109905585467737568 n^{4} + 14852840585693894845 n^{3} + 752619095443594533943 n^{2} + 16947387913246928559242 n + 143088766327097185851096\right) a{\left(n + 32 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(161426699054292668 n^{4} + 5252482297962464353 n^{3} + 62617357824753755230 n^{2} + 325330818680209958633 n + 623086556997783459072\right) a{\left(n + 6 \right)}}{40 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(474408101665695936 n^{4} + 56721906799189000642 n^{3} + 2540210323419533589477 n^{2} + 50497385327601946223315 n + 375952878720834816437982\right) a{\left(n + 29 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(742438770388733784 n^{4} + 89698100526323982521 n^{3} + 4055653962786953045439 n^{2} + 81319510782751927686076 n + 609959168103285776928468\right) a{\left(n + 30 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(1095182150231720140 n^{4} + 138658618965785370519 n^{3} + 6520871903734645989581 n^{2} + 135201396315248102130714 n + 1043929238413855618597404\right) a{\left(n + 28 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(2434236057501410494 n^{4} + 86821636393050800484 n^{3} + 1146455464407518473355 n^{2} + 6649674628467200823963 n + 14306486978573823934464\right) a{\left(n + 7 \right)}}{160 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(2661641226876060215 n^{4} + 105516395853346770897 n^{3} + 1553909654577380284381 n^{2} + 10079684462182074943584 n + 24309239015213590584330\right) a{\left(n + 8 \right)}}{80 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(3267432211933192831 n^{4} + 454271154724284504226 n^{3} + 20769133745246058246587 n^{2} + 393417653024990181936932 n + 2674503247666513338315138\right) a{\left(n + 22 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(6614535911630723275 n^{4} + 171385913342386364616 n^{3} + 406312218740865612701 n^{2} - 18404424759873275742414 n - 120363671111476080756624\right) a{\left(n + 11 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(6850572697097133572 n^{4} + 724124605053318632687 n^{3} + 28603108980924412402003 n^{2} + 500183768713666587529378 n + 3265546328358626478845016\right) a{\left(n + 27 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(8377681239583737686 n^{4} + 651207869789535996117 n^{3} + 18768470992765264520783 n^{2} + 237161978830493576296698 n + 1105167422620684615662384\right) a{\left(n + 21 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(11031482531251448377 n^{4} + 842369726409146808156 n^{3} + 24300757366847579320775 n^{2} + 314359672417914189719357 n + 1540803401192894500994300\right) a{\left(n + 19 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(11849150073753485332 n^{4} + 615321778587984032929 n^{3} + 11738850311502705569777 n^{2} + 97836660472256368847198 n + 301313855560040744395428\right) a{\left(n + 10 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(12766544158333342024 n^{4} + 1337346070708452471074 n^{3} + 52454696318682520561571 n^{2} + 912913282502345706496477 n + 5947502795228955649329582\right) a{\left(n + 26 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(15282750157919709658 n^{4} + 676333184657934871960 n^{3} + 11126371380199943137493 n^{2} + 80667192327398566316519 n + 217542936758213717129538\right) a{\left(n + 9 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(21171084178172935298 n^{4} + 2124283604158655523459 n^{3} + 79561476649751392136917 n^{2} + 1318824022674310040831946 n + 8166423963766963017349524\right) a{\left(n + 23 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(21541903879397745871 n^{4} + 1745185370080318340869 n^{3} + 53239846233118006589828 n^{2} + 725475520210016059567118 n + 3728900035826397129701310\right) a{\left(n + 20 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(29466424275665033319 n^{4} + 2951784723011174636842 n^{3} + 110714475255371122912767 n^{2} + 1842745061723509897002782 n + 11483405386731997642324344\right) a{\left(n + 24 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{3 \left(31832105202637336254 n^{4} + 2007106207682007427653 n^{3} + 47141568470015328133035 n^{2} + 488955292286476589806141 n + 1891097593940168132554214\right) a{\left(n + 17 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(48700903523066779442 n^{4} + 4981077020721326249687 n^{3} + 190809709526821970525053 n^{2} + 3244315214119336658097664 n + 20657104025398178364162108\right) a{\left(n + 25 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(82354825250731340918 n^{4} + 3724559200543369105071 n^{3} + 61864092715493979062977 n^{2} + 444099017570051223996048 n + 1148981115502212550117308\right) a{\left(n + 12 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(83806689305070961418 n^{4} + 4146849813159875182881 n^{3} + 76018177169995886009785 n^{2} + 610065169431970924628556 n + 1800766835236262656263396\right) a{\left(n + 13 \right)}}{320 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(117369035455142427782 n^{4} + 8090580405278867449051 n^{3} + 209636772756552719247901 n^{2} + 2425488190666036597495574 n + 10602819313907175957584772\right) a{\left(n + 18 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(240535095690493951244 n^{4} + 12617951173375116933505 n^{3} + 245122423418992007761243 n^{2} + 2083779039284955893950628 n + 6512469690773607512113932\right) a{\left(n + 14 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} - \frac{\left(253858678708561257756 n^{4} + 14907049415478806209699 n^{3} + 324196655836613006826573 n^{2} + 3087766005343445816352110 n + 10834551047862484950834216\right) a{\left(n + 16 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)} + \frac{\left(273720309364049278174 n^{4} + 15148953482930377420051 n^{3} + 310089612866803977082805 n^{2} + 2773299349124586372682304 n + 9101093245329040426087032\right) a{\left(n + 15 \right)}}{640 \left(n + 48\right) \left(n + 49\right) \left(n + 50\right) \left(2 n + 101\right)}, \quad n \geq 49\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 53 rules.
Finding the specification took 2661 seconds.
Copy 53 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_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0} \left(x \right)^{2} 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_{47}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{4}\! \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_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{9}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{0}\! \left(x \right) F_{29}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 54 rules.
Finding the specification took 4189 seconds.
Copy 54 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_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0} \left(x \right)^{2} 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_{48}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{4}\! \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_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{9}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \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_{0}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{16}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \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_{2}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{38}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \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_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{0}\! \left(x \right) F_{33}\! \left(x \right)\\
\end{align*}\)