Av(13254, 13524, 13542, 31254, 31524, 31542, 35124, 35142)
Counting Sequence
1, 1, 2, 6, 24, 112, 562, 2926, 15578, 84253, 461242, 2549908, 14211412, 79745508, 450081838, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -2\right) x^{3} F \left(x
\right)^{6}+6 x^{2} \left(x -1\right)^{2} F \left(x
\right)^{5}-x \left(13 x^{3}-6 x^{2}-9 x +6\right) F \left(x
\right)^{4}+\left(30 x^{3}-32 x^{2}+6 x +2\right) F \left(x
\right)^{3}+\left(-18 x^{3}+4 x^{2}+14 x -6\right) F \left(x
\right)^{2}+3 \left(3 x -2\right) \left(2 x -1\right) F \! \left(x \right)-\left(3 x -2\right) \left(2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2926\)
\(\displaystyle a \! \left(8\right) = 15578\)
\(\displaystyle a \! \left(9\right) = 84253\)
\(\displaystyle a \! \left(10\right) = 461242\)
\(\displaystyle a \! \left(11\right) = 2549908\)
\(\displaystyle a \! \left(12\right) = 14211412\)
\(\displaystyle a \! \left(13\right) = 79745508\)
\(\displaystyle a \! \left(14\right) = 450081838\)
\(\displaystyle a \! \left(15\right) = 2552946052\)
\(\displaystyle a \! \left(16\right) = 14543612544\)
\(\displaystyle a \! \left(17\right) = 83166912402\)
\(\displaystyle a \! \left(18\right) = 477181610838\)
\(\displaystyle a \! \left(19\right) = 2746062765924\)
\(\displaystyle a \! \left(20\right) = 15845094030930\)
\(\displaystyle a \! \left(21\right) = 91647577857421\)
\(\displaystyle a \! \left(22\right) = 531238621179010\)
\(\displaystyle a \! \left(23\right) = 3085424647445196\)
\(\displaystyle a \! \left(24\right) = 17952389365423098\)
\(\displaystyle a \! \left(25\right) = 104627601946549283\)
\(\displaystyle a \! \left(26\right) = 610702242526961350\)
\(\displaystyle a \! \left(27\right) = 3569613251565093942\)
\(\displaystyle a \! \left(28\right) = 20891809683421541276\)
\(\displaystyle a \! \left(29\right) = 122420428294303611536\)
\(\displaystyle a \! \left(30\right) = 718155588236979590482\)
\(\displaystyle a \! \left(31\right) = 4217329323643531070436\)
\(\displaystyle a \! \left(32\right) = 24790282381604530726796\)
\(\displaystyle a \! \left(33\right) = 145855848430514129466428\)
\(\displaystyle a \! \left(34\right) = 858895832897190722234320\)
\(\displaystyle a \! \left(35\right) = 5061851702512423832616392\)
\(\displaystyle a \! \left(36\right) = 29854548694843110801477498\)
\(\displaystyle a \! \left(37\right) = 176207892792195228625211173\)
\(\displaystyle a \! \left(38\right) = 1040727535291627765027158866\)
\(\displaystyle a \! \left(39\right) = 6150777306341935130665244364\)
\(\displaystyle a \! \left(40\right) = 36373900004700918791445493246\)
\(\displaystyle a \! \left(41\right) = 215230275755400347862499384975\)
\(\displaystyle a \! \left(42\right) = 1274260391927551993137009251588\)
\(\displaystyle a \! \left(43\right) = 7548191669338948327741305750870\)
\(\displaystyle a \! \left(44\right) = 44734949994803051787908178461814\)
\(\displaystyle a \! \left(45\right) = 265253064953627092177319142733213\)
\(\displaystyle a \! \left(46\right) = 1573526510128986223505621295962554\)
\(\displaystyle a \! \left(47\right) = 9338546874491412305633059106531384\)
\(\displaystyle a \! \left(48\right) = 55445725256792349985221809061689466\)
\(\displaystyle a \! \left(49\right) = 329331149245589501127848803281668011\)
\(\displaystyle a \! \left(50\right) = 1956890122620783815656649490334959856\)
\(\displaystyle a \! \left(51\right) = 11632211942661697554134060688482410438\)
\(\displaystyle a \! \left(52\right) = 69169421633029753786843838287958656882\)
\(\displaystyle a \! \left(n +53\right) = \frac{\left(2964226074114754 n^{5}+678782440049576265 n^{4}+62172719620912026385 n^{3}+2847276668471100104145 n^{2}+65195938985129275543471 n +597121914724689061066830\right) a \! \left(n +46\right)}{19968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(61744646960338 n^{5}+14436768906368450 n^{4}+1350164512400992055 n^{3}+63133496456645700850 n^{2}+1476012137167766923947 n +13802806500779982114120\right) a \! \left(n +47\right)}{6656 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(4840576901722 n^{5}+1155075720940965 n^{4}+110246736115408120 n^{3}+5261061618512995710 n^{2}+125526029485189744153 n +1197946250427714214080\right) a \! \left(n +48\right)}{9984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(102836271808 n^{5}+25032426750630 n^{4}+2437235239038175 n^{3}+118642460006622345 n^{2}+2887562270702235292 n +28110036146890642860\right) a \! \left(n +49\right)}{4992 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(60491874181498451088551 n^{5}+12086385348776061653084825 n^{4}+965982548873697796081549815 n^{3}+38603488129637769456249832325 n^{2}+771381454391805474633117347284 n +6165781719724849499820925679660\right) a \! \left(n +40\right)}{425984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(12204212824338224898004 n^{5}+2498105448866759660437305 n^{4}+204541478244972737096996110 n^{3}+8373992071722977580526595205 n^{2}+171421429826386869182112120316 n +1403685955212515874257764721100\right) a \! \left(n +41\right)}{638976 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(738803990063362177991 n^{5}+154834125310308351686295 n^{4}+12979856686280282813437760 n^{3}+544063633573815344312406375 n^{2}+11402678932437429760885901339 n +95594362990319952225963497700\right) a \! \left(n +42\right)}{319488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(39995229864389552783 n^{5}+8576820812345864306610 n^{4}+735709439995816060624745 n^{3}+31554327498765723671133765 n^{2}+676682393580312221782550087 n +5804634724339855689509270520\right) a \! \left(n +43\right)}{159744 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(640216836621929027 n^{5}+140404007595939901175 n^{4}+12316571587482644205290 n^{3}+540217651612087544771935 n^{2}+11847234369207933201012023 n +103926216816763797805898310\right) a \! \left(n +44\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{5 \left(10801437280205748 n^{5}+2421203002437945951 n^{4}+217087503293684098932 n^{3}+9732020711730649980569 n^{2}+218140551758273639288028 n +1955805032280616673187256\right) a \! \left(n +45\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(4980926822738138989463879 n^{5}+946395902053748680321081545 n^{4}+71931052110727477918217577095 n^{3}+2733707035281782379030896317045 n^{2}+51949410131708452619664271037996 n +394905725467275371354445081405560\right) a \! \left(n +38\right)}{851968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(1221205355320307096642029 n^{5}+238019148633223122808695330 n^{4}+18557193379886376724590262180 n^{3}+723437370082824442793113991115 n^{2}+14101939907118397397123708632996 n +109960588352970760479617148961080\right) a \! \left(n +39\right)}{1277952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(3997616203049382749690509814 n^{5}+700762401858985513085390974215 n^{4}+49139633465754539358858176653910 n^{3}+1723041665643482631224845502105295 n^{2}+30210900781555391185047127424767346 n +211897991022659425101638674184771460\right) a \! \left(n +35\right)}{5111808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(426300824490033205598141276 n^{5}+76818094495927912478670200145 n^{4}+5537310306447940022639203279370 n^{3}+199587541957676447457716490652815 n^{2}+3597232100532603397666352918885054 n +25935530978775306142636860127769340\right) a \! \left(n +36\right)}{2555904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(83424235297909456326686417 n^{5}+15441871812003582792778793685 n^{4}+1143384007198957022952818654300 n^{3}+42333087214332525015524119381560 n^{2}+783726606246080931507550129856978 n +5804130819831183626833544804504340\right) a \! \left(n +37\right)}{2555904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(1001160399155005462104831067702 n^{5}+160803063610740151857288737380395 n^{4}+10332054877380071175788192957307685 n^{3}+331964967826892556340455808304548395 n^{2}+5333500624274483059185564526897055713 n +34279802452237502476820241511177299330\right) a \! \left(n +32\right)}{20447232 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(45599458775363479364624135609 n^{5}+7546874184190350157394809037765 n^{4}+499657939151107585091301919384675 n^{3}+16542004975675220730935262171460180 n^{2}+273851128505414139745911536212256301 n +1813607341931469527667017542863658470\right) a \! \left(n +33\right)}{3407872 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(5744108747435943293983946759 n^{5}+978777671158577329638689245200 n^{4}+66717581349053207031194663717135 n^{3}+2274064338924828613099943852670110 n^{2}+38759015997310275035333439477692336 n +264266163947401207758608756915334460\right) a \! \left(n +34\right)}{1703936 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(121386713111693902563306921240977 n^{5}+17722550706055345238962381017972240 n^{4}+1035128385835469440517946789303609560 n^{3}+30233366809768963083616861585062954435 n^{2}+441574941905682598206470802292782655988 n +2580115603933950686797583009591743239720\right) a \! \left(n +29\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(42154354754271710846410412485241 n^{5}+6359611580026798594941506241128715 n^{4}+383820128864369617723825287605056325 n^{3}+11583647702276767771758709749217860495 n^{2}+174817428959306929022141120643254067344 n +1055448856858205078440190407192839338160\right) a \! \left(n +30\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(2253610412089793867700669317327 n^{5}+350970375151194209563774730820485 n^{4}+21865900919342335697592877943629835 n^{3}+681210025221625408886348727199485775 n^{2}+10612380254657533729377727107013881698 n +66138448902261130014612374103318657600\right) a \! \left(n +31\right)}{13631488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(3174475819630998406401720075140686 n^{5}+432667838533925293378298642290785975 n^{4}+23591667672431090990693977008155958040 n^{3}+643273003226808500749444673797840177875 n^{2}+8771336629918967031662335488061105303204 n +47847693987930032564984725555346969988480\right) a \! \left(n +27\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(322920507115274101908627663550606 n^{5}+45578451343330561283928230517628005 n^{4}+2573597060249963159831074701989725705 n^{3}+72669121273986999571046118114718835910 n^{2}+1026098055151375759956524962132964941924 n +5796269305884562715114770575097668334260\right) a \! \left(n +28\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{3 \left(2894758972633232694210913121693417 n^{5}+338586660241393969060025388972014530 n^{4}+15844148961328404693803409742082151305 n^{3}+370783083443941216176109954355643398740 n^{2}+4339332643266941915293748640819262340228 n +20317392128543291400745217614669276978400\right) a \! \left(n +23\right)}{54525952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{5 \left(3895060665207283878685777651443603 n^{5}+474379059514050818792930965599922147 n^{4}+23113888324559966567922693817690837979 n^{3}+563206097135591169006081045912017023109 n^{2}+6862920848030820419413960793385998856730 n +33457097289212360901785862590126938684560\right) a \! \left(n +24\right)}{218103808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{5 \left(3020695856446882481941825839256466 n^{5}+382478223189390694622873964122388201 n^{4}+19374810077551934114378773947980318132 n^{3}+490806267380405451117108391623425266131 n^{2}+6217637454431896280515989291645675893438 n +31511853752707252912237295161349919760560\right) a \! \left(n +25\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(14412135337430962080853283034403759 n^{5}+1894537151853555961098258079450539855 n^{4}+99633185784153482844986017636222350085 n^{3}+2620246069360663109325518365574638683105 n^{2}+34460237386668943751698823119318951692796 n +181310691927760799851598394989572170864960\right) a \! \left(n +26\right)}{654311424 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(129508842466323188919494233539370094 n^{5}+13899797328307430124882756317487762645 n^{4}+596857296489345740607374905165411141040 n^{3}+12817249626116432526669581984023100268675 n^{2}+137651379022079006673865467050889179297426 n +591448005887673661014839420156402441457000\right) a \! \left(n +21\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(28542745382375085431450900963764559 n^{5}+3200920262562707790994368698870398425 n^{4}+143615467662413070725075779993376399425 n^{3}+3222438807393508466746167844084135736095 n^{2}+36159707331631228958970636203926111188696 n +162334707130029598760290598830997778233120\right) a \! \left(n +22\right)}{109051904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(268488150331272093523701361145946421 n^{5}+24936864825807895275206941234407235035 n^{4}+926678083927739174631054992442801774415 n^{3}+17222366664039173243451108626486759059605 n^{2}+160077384837317438621177550735605326471084 n +595287008047973853404097036783564825635760\right) a \! \left(n +18\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(229703375767717602914117258369640779 n^{5}+22440945767082511055784444960131128575 n^{4}+877161455810158986244522277314439528735 n^{3}+17147082713761227062562374865565643894145 n^{2}+167637326949757919430025200497176853027366 n +655704583114223989484368809924242913316800\right) a \! \left(n +19\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{3 \left(40008950201067261521823088129277549 n^{5}+4101354914474141230381346686610473645 n^{4}+168211759808401872879455340351641438785 n^{3}+3450267466528279426975657346658041332475 n^{2}+35392802364371058407002957446439666940386 n +145254993428137731711275746069704374793480\right) a \! \left(n +20\right)}{218103808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(20665954943608266114592965382998874 n^{5}+1620379355100505533738813515860688740 n^{4}+50835143321901543661042936687119288725 n^{3}+797624462319855721772097865718212029125 n^{2}+6259081059630006326367644559119586753246 n +19650825177174914256132580386856199156640\right) a \! \left(n +15\right)}{27262976 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(139835882989736713831170238644495983 n^{5}+11639522404152992682178303492326381585 n^{4}+387643785782000952198749248438954468475 n^{3}+6456764271257391402648840774216496594635 n^{2}+53786648104012600768314931814057185844282 n +179264472828115013024748907696255805514600\right) a \! \left(n +16\right)}{163577856 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(47819639587855549714289836266778701 n^{5}+4210995156350317849766991270277211815 n^{4}+148367586251873992919181158868323342665 n^{3}+2614413717436512265089958593941743771235 n^{2}+23040157586838916719883422592294210187624 n +81237589299772468086753643712156309603640\right) a \! \left(n +17\right)}{54525952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{9 \left(109767025107387071519025139058566 n^{5}+7010092507718169123182586314104995 n^{4}+179132738181925042381606030173713485 n^{3}+2289359829126236855976632239101178815 n^{2}+14632530408827464109315148793917916204 n +37416190627289619489521566955453416530\right) a \! \left(n +12\right)}{3407872 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{9 \left(335533868506189267623718939340814 n^{5}+23059173328424348146248618725040080 n^{4}+634083299108884058987760681424113995 n^{3}+8720430785885390192001175387189210315 n^{2}+59979414676317339691647399670365236761 n +165048921534812893328676247154969087250\right) a \! \left(n +13\right)}{6815744 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{3 \left(2769902621138651756537903128445562 n^{5}+203785110616870281766247207187053215 n^{4}+5998894305171277959624243334056730310 n^{3}+88319914672975024362838075664780813570 n^{2}+650312835081847439857291530506269123483 n +1915753672883899179447291288878429412030\right) a \! \left(n +14\right)}{13631488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{243 \left(72134159081467067073911813736 n^{5}+3547330245280331933494203338670 n^{4}+69802250685564818336829893969275 n^{3}+686915574427836641504127375410565 n^{2}+3380284388879906227929749769356564 n +6653654578990292836282905316551540\right) a \! \left(n +9\right)}{425984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{405 \left(186893432743936759593936118206 n^{5}+10109492529176618231526209147493 n^{4}+218811922967466966268169393828272 n^{3}+2368595080673553909399632262136803 n^{2}+12821863264978274145001366018214345 n +27765464511948481444385054829888102\right) a \! \left(n +10\right)}{851968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{27 \left(10726593455466535290929982109102 n^{5}+632730433627131715101480466904270 n^{4}+14934092379285171913487775342234735 n^{3}+176288337068020129980220908722393965 n^{2}+1040697396065807549518309522036122623 n +2457781791606866190780960287764551690\right) a \! \left(n +11\right)}{1703936 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{6561 \left(7400338090551525791762031 n^{5}+253916618626066351458595650 n^{4}+3485362218759881375681817775 n^{3}+23916900786808807254290383860 n^{2}+82023731073648492548866209584 n +112441270048148598225772075560\right) a \! \left(n +6\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{2187 \left(145010734573838746452107832 n^{5}+5696458119127866748135262115 n^{4}+89534205220452713994829367690 n^{3}+703670553556252027995120125355 n^{2}+2764749352157571785128305632548 n +4343619698566060369163997700200\right) a \! \left(n +7\right)}{53248 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{729 \left(4898375758437829734795371061 n^{5}+216703679226314879005088759610 n^{4}+3836030277163816809072091220480 n^{3}+33957886050308682978433487278905 n^{2}+150305551104409447686340280005534 n +266078888965805590675147273259520\right) a \! \left(n +8\right)}{212992 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{885735 \left(16517366109227422132 n^{5}+321082279477027080543 n^{4}+2489005916158610100332 n^{3}+9609311188394108115297 n^{2}+18460990765608112501518 n +14108382958336155825588\right) a \! \left(n +3\right)}{832 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{295245 \left(1150412606997915320121 n^{5}+28037477364396616733562 n^{4}+273039429949924879438663 n^{3}+1327303891403587373902782 n^{2}+3219106047458731530165140 n +3114512879225224771028952\right) a \! \left(n +4\right)}{3328 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{19683 \left(160216024156245105985422 n^{5}+4700057503083087753117765 n^{4}+55139660165765538576179590 n^{3}+323237995233961918467534435 n^{2}+946475813831042922249250508 n +1107025595903177960981785380\right) a \! \left(n +5\right)}{6656 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{2657205 \left(n +2\right) \left(356986057911353707 n^{4}+4482989967309818322 n^{3}+21046912309082360507 n^{2}+43716688926478485480 n +33850197454932964854\right) a \! \left(n +2\right)}{416 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{2176250895 \left(2 n +3\right) \left(n +2\right) \left(n +1\right) \left(364321627307 n^{2}+1910123893888 n +2519090134676\right) a \! \left(n +1\right)}{8 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{8660943380034366525 n \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{4 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(13558 n^{2}+1379427 n +35081610\right) a \! \left(n +52\right)}{52 \left(n +54\right) \left(2 n +105\right)}+\frac{\left(284098108 n^{4}+56025496602 n^{3}+4143089481288 n^{2}+136166567340327 n +1678178564080910\right) a \! \left(n +50\right)}{416 \left(n +52\right) \left(n +53\right) \left(n +54\right) \left(2 n +105\right)}-\frac{\left(3443408 n^{3}+516436636 n^{2}+25817459551 n +430207453430\right) a \! \left(n +51\right)}{208 \left(n +53\right) \left(n +54\right) \left(2 n +105\right)}, \quad n \geq 53\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2926\)
\(\displaystyle a \! \left(8\right) = 15578\)
\(\displaystyle a \! \left(9\right) = 84253\)
\(\displaystyle a \! \left(10\right) = 461242\)
\(\displaystyle a \! \left(11\right) = 2549908\)
\(\displaystyle a \! \left(12\right) = 14211412\)
\(\displaystyle a \! \left(13\right) = 79745508\)
\(\displaystyle a \! \left(14\right) = 450081838\)
\(\displaystyle a \! \left(15\right) = 2552946052\)
\(\displaystyle a \! \left(16\right) = 14543612544\)
\(\displaystyle a \! \left(17\right) = 83166912402\)
\(\displaystyle a \! \left(18\right) = 477181610838\)
\(\displaystyle a \! \left(19\right) = 2746062765924\)
\(\displaystyle a \! \left(20\right) = 15845094030930\)
\(\displaystyle a \! \left(21\right) = 91647577857421\)
\(\displaystyle a \! \left(22\right) = 531238621179010\)
\(\displaystyle a \! \left(23\right) = 3085424647445196\)
\(\displaystyle a \! \left(24\right) = 17952389365423098\)
\(\displaystyle a \! \left(25\right) = 104627601946549283\)
\(\displaystyle a \! \left(26\right) = 610702242526961350\)
\(\displaystyle a \! \left(27\right) = 3569613251565093942\)
\(\displaystyle a \! \left(28\right) = 20891809683421541276\)
\(\displaystyle a \! \left(29\right) = 122420428294303611536\)
\(\displaystyle a \! \left(30\right) = 718155588236979590482\)
\(\displaystyle a \! \left(31\right) = 4217329323643531070436\)
\(\displaystyle a \! \left(32\right) = 24790282381604530726796\)
\(\displaystyle a \! \left(33\right) = 145855848430514129466428\)
\(\displaystyle a \! \left(34\right) = 858895832897190722234320\)
\(\displaystyle a \! \left(35\right) = 5061851702512423832616392\)
\(\displaystyle a \! \left(36\right) = 29854548694843110801477498\)
\(\displaystyle a \! \left(37\right) = 176207892792195228625211173\)
\(\displaystyle a \! \left(38\right) = 1040727535291627765027158866\)
\(\displaystyle a \! \left(39\right) = 6150777306341935130665244364\)
\(\displaystyle a \! \left(40\right) = 36373900004700918791445493246\)
\(\displaystyle a \! \left(41\right) = 215230275755400347862499384975\)
\(\displaystyle a \! \left(42\right) = 1274260391927551993137009251588\)
\(\displaystyle a \! \left(43\right) = 7548191669338948327741305750870\)
\(\displaystyle a \! \left(44\right) = 44734949994803051787908178461814\)
\(\displaystyle a \! \left(45\right) = 265253064953627092177319142733213\)
\(\displaystyle a \! \left(46\right) = 1573526510128986223505621295962554\)
\(\displaystyle a \! \left(47\right) = 9338546874491412305633059106531384\)
\(\displaystyle a \! \left(48\right) = 55445725256792349985221809061689466\)
\(\displaystyle a \! \left(49\right) = 329331149245589501127848803281668011\)
\(\displaystyle a \! \left(50\right) = 1956890122620783815656649490334959856\)
\(\displaystyle a \! \left(51\right) = 11632211942661697554134060688482410438\)
\(\displaystyle a \! \left(52\right) = 69169421633029753786843838287958656882\)
\(\displaystyle a \! \left(n +53\right) = \frac{\left(2964226074114754 n^{5}+678782440049576265 n^{4}+62172719620912026385 n^{3}+2847276668471100104145 n^{2}+65195938985129275543471 n +597121914724689061066830\right) a \! \left(n +46\right)}{19968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(61744646960338 n^{5}+14436768906368450 n^{4}+1350164512400992055 n^{3}+63133496456645700850 n^{2}+1476012137167766923947 n +13802806500779982114120\right) a \! \left(n +47\right)}{6656 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(4840576901722 n^{5}+1155075720940965 n^{4}+110246736115408120 n^{3}+5261061618512995710 n^{2}+125526029485189744153 n +1197946250427714214080\right) a \! \left(n +48\right)}{9984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(102836271808 n^{5}+25032426750630 n^{4}+2437235239038175 n^{3}+118642460006622345 n^{2}+2887562270702235292 n +28110036146890642860\right) a \! \left(n +49\right)}{4992 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(60491874181498451088551 n^{5}+12086385348776061653084825 n^{4}+965982548873697796081549815 n^{3}+38603488129637769456249832325 n^{2}+771381454391805474633117347284 n +6165781719724849499820925679660\right) a \! \left(n +40\right)}{425984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(12204212824338224898004 n^{5}+2498105448866759660437305 n^{4}+204541478244972737096996110 n^{3}+8373992071722977580526595205 n^{2}+171421429826386869182112120316 n +1403685955212515874257764721100\right) a \! \left(n +41\right)}{638976 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(738803990063362177991 n^{5}+154834125310308351686295 n^{4}+12979856686280282813437760 n^{3}+544063633573815344312406375 n^{2}+11402678932437429760885901339 n +95594362990319952225963497700\right) a \! \left(n +42\right)}{319488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(39995229864389552783 n^{5}+8576820812345864306610 n^{4}+735709439995816060624745 n^{3}+31554327498765723671133765 n^{2}+676682393580312221782550087 n +5804634724339855689509270520\right) a \! \left(n +43\right)}{159744 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(640216836621929027 n^{5}+140404007595939901175 n^{4}+12316571587482644205290 n^{3}+540217651612087544771935 n^{2}+11847234369207933201012023 n +103926216816763797805898310\right) a \! \left(n +44\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{5 \left(10801437280205748 n^{5}+2421203002437945951 n^{4}+217087503293684098932 n^{3}+9732020711730649980569 n^{2}+218140551758273639288028 n +1955805032280616673187256\right) a \! \left(n +45\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(4980926822738138989463879 n^{5}+946395902053748680321081545 n^{4}+71931052110727477918217577095 n^{3}+2733707035281782379030896317045 n^{2}+51949410131708452619664271037996 n +394905725467275371354445081405560\right) a \! \left(n +38\right)}{851968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(1221205355320307096642029 n^{5}+238019148633223122808695330 n^{4}+18557193379886376724590262180 n^{3}+723437370082824442793113991115 n^{2}+14101939907118397397123708632996 n +109960588352970760479617148961080\right) a \! \left(n +39\right)}{1277952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(3997616203049382749690509814 n^{5}+700762401858985513085390974215 n^{4}+49139633465754539358858176653910 n^{3}+1723041665643482631224845502105295 n^{2}+30210900781555391185047127424767346 n +211897991022659425101638674184771460\right) a \! \left(n +35\right)}{5111808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(426300824490033205598141276 n^{5}+76818094495927912478670200145 n^{4}+5537310306447940022639203279370 n^{3}+199587541957676447457716490652815 n^{2}+3597232100532603397666352918885054 n +25935530978775306142636860127769340\right) a \! \left(n +36\right)}{2555904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(83424235297909456326686417 n^{5}+15441871812003582792778793685 n^{4}+1143384007198957022952818654300 n^{3}+42333087214332525015524119381560 n^{2}+783726606246080931507550129856978 n +5804130819831183626833544804504340\right) a \! \left(n +37\right)}{2555904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(1001160399155005462104831067702 n^{5}+160803063610740151857288737380395 n^{4}+10332054877380071175788192957307685 n^{3}+331964967826892556340455808304548395 n^{2}+5333500624274483059185564526897055713 n +34279802452237502476820241511177299330\right) a \! \left(n +32\right)}{20447232 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(45599458775363479364624135609 n^{5}+7546874184190350157394809037765 n^{4}+499657939151107585091301919384675 n^{3}+16542004975675220730935262171460180 n^{2}+273851128505414139745911536212256301 n +1813607341931469527667017542863658470\right) a \! \left(n +33\right)}{3407872 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(5744108747435943293983946759 n^{5}+978777671158577329638689245200 n^{4}+66717581349053207031194663717135 n^{3}+2274064338924828613099943852670110 n^{2}+38759015997310275035333439477692336 n +264266163947401207758608756915334460\right) a \! \left(n +34\right)}{1703936 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(121386713111693902563306921240977 n^{5}+17722550706055345238962381017972240 n^{4}+1035128385835469440517946789303609560 n^{3}+30233366809768963083616861585062954435 n^{2}+441574941905682598206470802292782655988 n +2580115603933950686797583009591743239720\right) a \! \left(n +29\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(42154354754271710846410412485241 n^{5}+6359611580026798594941506241128715 n^{4}+383820128864369617723825287605056325 n^{3}+11583647702276767771758709749217860495 n^{2}+174817428959306929022141120643254067344 n +1055448856858205078440190407192839338160\right) a \! \left(n +30\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(2253610412089793867700669317327 n^{5}+350970375151194209563774730820485 n^{4}+21865900919342335697592877943629835 n^{3}+681210025221625408886348727199485775 n^{2}+10612380254657533729377727107013881698 n +66138448902261130014612374103318657600\right) a \! \left(n +31\right)}{13631488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(3174475819630998406401720075140686 n^{5}+432667838533925293378298642290785975 n^{4}+23591667672431090990693977008155958040 n^{3}+643273003226808500749444673797840177875 n^{2}+8771336629918967031662335488061105303204 n +47847693987930032564984725555346969988480\right) a \! \left(n +27\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(322920507115274101908627663550606 n^{5}+45578451343330561283928230517628005 n^{4}+2573597060249963159831074701989725705 n^{3}+72669121273986999571046118114718835910 n^{2}+1026098055151375759956524962132964941924 n +5796269305884562715114770575097668334260\right) a \! \left(n +28\right)}{81788928 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{3 \left(2894758972633232694210913121693417 n^{5}+338586660241393969060025388972014530 n^{4}+15844148961328404693803409742082151305 n^{3}+370783083443941216176109954355643398740 n^{2}+4339332643266941915293748640819262340228 n +20317392128543291400745217614669276978400\right) a \! \left(n +23\right)}{54525952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{5 \left(3895060665207283878685777651443603 n^{5}+474379059514050818792930965599922147 n^{4}+23113888324559966567922693817690837979 n^{3}+563206097135591169006081045912017023109 n^{2}+6862920848030820419413960793385998856730 n +33457097289212360901785862590126938684560\right) a \! \left(n +24\right)}{218103808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{5 \left(3020695856446882481941825839256466 n^{5}+382478223189390694622873964122388201 n^{4}+19374810077551934114378773947980318132 n^{3}+490806267380405451117108391623425266131 n^{2}+6217637454431896280515989291645675893438 n +31511853752707252912237295161349919760560\right) a \! \left(n +25\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(14412135337430962080853283034403759 n^{5}+1894537151853555961098258079450539855 n^{4}+99633185784153482844986017636222350085 n^{3}+2620246069360663109325518365574638683105 n^{2}+34460237386668943751698823119318951692796 n +181310691927760799851598394989572170864960\right) a \! \left(n +26\right)}{654311424 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(129508842466323188919494233539370094 n^{5}+13899797328307430124882756317487762645 n^{4}+596857296489345740607374905165411141040 n^{3}+12817249626116432526669581984023100268675 n^{2}+137651379022079006673865467050889179297426 n +591448005887673661014839420156402441457000\right) a \! \left(n +21\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(28542745382375085431450900963764559 n^{5}+3200920262562707790994368698870398425 n^{4}+143615467662413070725075779993376399425 n^{3}+3222438807393508466746167844084135736095 n^{2}+36159707331631228958970636203926111188696 n +162334707130029598760290598830997778233120\right) a \! \left(n +22\right)}{109051904 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(268488150331272093523701361145946421 n^{5}+24936864825807895275206941234407235035 n^{4}+926678083927739174631054992442801774415 n^{3}+17222366664039173243451108626486759059605 n^{2}+160077384837317438621177550735605326471084 n +595287008047973853404097036783564825635760\right) a \! \left(n +18\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(229703375767717602914117258369640779 n^{5}+22440945767082511055784444960131128575 n^{4}+877161455810158986244522277314439528735 n^{3}+17147082713761227062562374865565643894145 n^{2}+167637326949757919430025200497176853027366 n +655704583114223989484368809924242913316800\right) a \! \left(n +19\right)}{327155712 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{3 \left(40008950201067261521823088129277549 n^{5}+4101354914474141230381346686610473645 n^{4}+168211759808401872879455340351641438785 n^{3}+3450267466528279426975657346658041332475 n^{2}+35392802364371058407002957446439666940386 n +145254993428137731711275746069704374793480\right) a \! \left(n +20\right)}{218103808 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(20665954943608266114592965382998874 n^{5}+1620379355100505533738813515860688740 n^{4}+50835143321901543661042936687119288725 n^{3}+797624462319855721772097865718212029125 n^{2}+6259081059630006326367644559119586753246 n +19650825177174914256132580386856199156640\right) a \! \left(n +15\right)}{27262976 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(139835882989736713831170238644495983 n^{5}+11639522404152992682178303492326381585 n^{4}+387643785782000952198749248438954468475 n^{3}+6456764271257391402648840774216496594635 n^{2}+53786648104012600768314931814057185844282 n +179264472828115013024748907696255805514600\right) a \! \left(n +16\right)}{163577856 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{\left(47819639587855549714289836266778701 n^{5}+4210995156350317849766991270277211815 n^{4}+148367586251873992919181158868323342665 n^{3}+2614413717436512265089958593941743771235 n^{2}+23040157586838916719883422592294210187624 n +81237589299772468086753643712156309603640\right) a \! \left(n +17\right)}{54525952 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{9 \left(109767025107387071519025139058566 n^{5}+7010092507718169123182586314104995 n^{4}+179132738181925042381606030173713485 n^{3}+2289359829126236855976632239101178815 n^{2}+14632530408827464109315148793917916204 n +37416190627289619489521566955453416530\right) a \! \left(n +12\right)}{3407872 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{9 \left(335533868506189267623718939340814 n^{5}+23059173328424348146248618725040080 n^{4}+634083299108884058987760681424113995 n^{3}+8720430785885390192001175387189210315 n^{2}+59979414676317339691647399670365236761 n +165048921534812893328676247154969087250\right) a \! \left(n +13\right)}{6815744 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{3 \left(2769902621138651756537903128445562 n^{5}+203785110616870281766247207187053215 n^{4}+5998894305171277959624243334056730310 n^{3}+88319914672975024362838075664780813570 n^{2}+650312835081847439857291530506269123483 n +1915753672883899179447291288878429412030\right) a \! \left(n +14\right)}{13631488 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{243 \left(72134159081467067073911813736 n^{5}+3547330245280331933494203338670 n^{4}+69802250685564818336829893969275 n^{3}+686915574427836641504127375410565 n^{2}+3380284388879906227929749769356564 n +6653654578990292836282905316551540\right) a \! \left(n +9\right)}{425984 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{405 \left(186893432743936759593936118206 n^{5}+10109492529176618231526209147493 n^{4}+218811922967466966268169393828272 n^{3}+2368595080673553909399632262136803 n^{2}+12821863264978274145001366018214345 n +27765464511948481444385054829888102\right) a \! \left(n +10\right)}{851968 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{27 \left(10726593455466535290929982109102 n^{5}+632730433627131715101480466904270 n^{4}+14934092379285171913487775342234735 n^{3}+176288337068020129980220908722393965 n^{2}+1040697396065807549518309522036122623 n +2457781791606866190780960287764551690\right) a \! \left(n +11\right)}{1703936 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{6561 \left(7400338090551525791762031 n^{5}+253916618626066351458595650 n^{4}+3485362218759881375681817775 n^{3}+23916900786808807254290383860 n^{2}+82023731073648492548866209584 n +112441270048148598225772075560\right) a \! \left(n +6\right)}{26624 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{2187 \left(145010734573838746452107832 n^{5}+5696458119127866748135262115 n^{4}+89534205220452713994829367690 n^{3}+703670553556252027995120125355 n^{2}+2764749352157571785128305632548 n +4343619698566060369163997700200\right) a \! \left(n +7\right)}{53248 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{729 \left(4898375758437829734795371061 n^{5}+216703679226314879005088759610 n^{4}+3836030277163816809072091220480 n^{3}+33957886050308682978433487278905 n^{2}+150305551104409447686340280005534 n +266078888965805590675147273259520\right) a \! \left(n +8\right)}{212992 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{885735 \left(16517366109227422132 n^{5}+321082279477027080543 n^{4}+2489005916158610100332 n^{3}+9609311188394108115297 n^{2}+18460990765608112501518 n +14108382958336155825588\right) a \! \left(n +3\right)}{832 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{295245 \left(1150412606997915320121 n^{5}+28037477364396616733562 n^{4}+273039429949924879438663 n^{3}+1327303891403587373902782 n^{2}+3219106047458731530165140 n +3114512879225224771028952\right) a \! \left(n +4\right)}{3328 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{19683 \left(160216024156245105985422 n^{5}+4700057503083087753117765 n^{4}+55139660165765538576179590 n^{3}+323237995233961918467534435 n^{2}+946475813831042922249250508 n +1107025595903177960981785380\right) a \! \left(n +5\right)}{6656 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{2657205 \left(n +2\right) \left(356986057911353707 n^{4}+4482989967309818322 n^{3}+21046912309082360507 n^{2}+43716688926478485480 n +33850197454932964854\right) a \! \left(n +2\right)}{416 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}-\frac{2176250895 \left(2 n +3\right) \left(n +2\right) \left(n +1\right) \left(364321627307 n^{2}+1910123893888 n +2519090134676\right) a \! \left(n +1\right)}{8 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{8660943380034366525 n \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{4 \left(2 n +105\right) \left(n +54\right) \left(n +53\right) \left(n +52\right) \left(n +51\right)}+\frac{\left(13558 n^{2}+1379427 n +35081610\right) a \! \left(n +52\right)}{52 \left(n +54\right) \left(2 n +105\right)}+\frac{\left(284098108 n^{4}+56025496602 n^{3}+4143089481288 n^{2}+136166567340327 n +1678178564080910\right) a \! \left(n +50\right)}{416 \left(n +52\right) \left(n +53\right) \left(n +54\right) \left(2 n +105\right)}-\frac{\left(3443408 n^{3}+516436636 n^{2}+25817459551 n +430207453430\right) a \! \left(n +51\right)}{208 \left(n +53\right) \left(n +54\right) \left(2 n +105\right)}, \quad n \geq 53\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 102 rules.
Found on January 23, 2022.Finding the specification took 255 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{11}\! \left(x \right) F_{48}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{48}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{28}\! \left(x \right) F_{44}\! \left(x , y\right) F_{47}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{78}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{28}\! \left(x \right) F_{44}\! \left(x , y\right) F_{48}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 0\\
F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{11}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= y x\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{11}\! \left(x \right) F_{49}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{55} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{11}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{65}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{11}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{11}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{11}\! \left(x \right) F_{59}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{11}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{31}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{11}\! \left(x \right) F_{67}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{11}\! \left(x \right) F_{55}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{11}\! \left(x \right) F_{49}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{28}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{81}\! \left(x \right) &= -F_{87}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= \frac{F_{91}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{2}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{11}\! \left(x \right) F_{96}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{96}\! \left(x \right) &= \frac{F_{97}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{97}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{58}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 26 rules.
Found on January 22, 2022.Finding the specification took 47 seconds.
Copy 26 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_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , 1, y\right)\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y z , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)+F_{24}\! \left(x , y , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , z\right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x , y , z\right) &= \frac{y F_{14}\! \left(x , y\right)-z F_{14}\! \left(x , z\right)}{-z +y}\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x , y\right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= \frac{y F_{20}\! \left(x , y , 1\right)-z F_{20}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\
F_{20}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y , z\right)+F_{22}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , y z \right)\\
F_{22}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , y z \right)\\
F_{23}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{20}\! \left(x , y , z\right)\\
F_{24}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= -\frac{z F_{10}\! \left(x , 1, z\right)-y F_{10}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
\end{align*}\)