1, 1, 2, 6, 24, 111, 547, 2763, 14136, 73095, 381979, 2016726, 10748797, 57773635, 312832005, ...

\(\displaystyle x^{6} \left(x^{2}-5 x +1\right) F \left(x \right)^{5}-x^{3} \left(x^{5}-6 x^{4}-8 x^{3}+6 x^{2}-4 x +1\right) F \left(x \right)^{4}-x^{2} \left(x^{6}-3 x^{5}+15 x^{4}+5 x^{3}-17 x^{2}+13 x -3\right) F \left(x \right)^{3}+x \left(x^{6}-3 x^{5}+11 x^{4}+2 x^{3}-19 x^{2}+16 x -4\right) F \left(x \right)^{2}+\left(3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{3} = 0\)

\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 111\)
\(\displaystyle a(6) = 547\)
\(\displaystyle a(7) = 2763\)
\(\displaystyle a(8) = 14136\)
\(\displaystyle a(9) = 73095\)
\(\displaystyle a(10) = 381979\)
\(\displaystyle a(11) = 2016726\)
\(\displaystyle a(12) = 10748797\)
\(\displaystyle a(13) = 57773635\)
\(\displaystyle a(14) = 312832005\)
\(\displaystyle a(15) = 1704943887\)
\(\displaystyle a(16) = 9345301332\)
\(\displaystyle a(17) = 51485226705\)
\(\displaystyle a(18) = 284934205545\)
\(\displaystyle a(19) = 1583365758827\)
\(\displaystyle a(20) = 8831267596639\)
\(\displaystyle a(21) = 49422355171295\)
\(\displaystyle a(22) = 277430934929097\)
\(\displaystyle a(23) = 1561729128884587\)
\(\displaystyle a(24) = 8814092494856060\)
\(\displaystyle a(25) = 49863596330059678\)
\(\displaystyle a(26) = 282713410211634108\)
\(\displaystyle a(27) = 1606190529798914622\)
\(\displaystyle a(28) = 9142684714692688359\)
\(\displaystyle a(29) = 52133977403548017045\)
\(\displaystyle a(30) = 297774992553723072381\)
\(\displaystyle a(31) = 1703453880096747753300\)
\(\displaystyle a(32) = 9759016163313880404275\)
\(\displaystyle a(33) = 55985758798192874914613\)
\(\displaystyle a(34) = 321595968254669013943962\)
\(\displaystyle a(35) = 1849582143995832444646784\)
\(\displaystyle a(36) = 10649708801964472021456211\)
\(\displaystyle a(37) = 61386984554239297103941437\)
\(\displaystyle a(38) = 354213221958638158594777291\)
\(\displaystyle a(39) = 2045881302914775833618683039\)
\(\displaystyle a(40) = 11827753535223978135506945457\)
\(\displaystyle a(41) = 68440118157342301796189782001\)
\(\displaystyle a(42) = 396358104114384319823425717033\)
\(\displaystyle a(43) = 2297292757353122201149847606764\)
\(\displaystyle a(44) = 13325416445375359100970552990367\)
\(\displaystyle a(45) = 77351075414710052845045507464628\)
\(\displaystyle a(46) = 449323630905732096724122808418289\)
\(\displaystyle a(47) = 2611840117178392598629959811735023\)
\(\displaystyle a(48) = 15192044542832994871553591521857584\)
\(\displaystyle a(49) = 88421277597701781554724645033037251\)
\(\displaystyle a(50) = 514941104502036542402855420752429111\)
\(\displaystyle a(51) = 3000602458376078895180467454274411443\)
\(\displaystyle a(52) = 17494437157246452668525170388279242408\)
\(\displaystyle a(53) = 102052376187220480325189755641781843434\)
\(\displaystyle a(54) = 595620193496097479518876967727772456069\)
\(\displaystyle a(55) = 3478009959300069523902797881486201105282\)
\(\displaystyle a(56) = 20318878853297277090684004933931166659348\)
\(\displaystyle a(57) = 118759665103604467302433237709787647228419\)
\(\displaystyle a(58) = 694435080696395604967386562487281331796547\)
\(\displaystyle a(59) = 4062387142867036121287445171317199852783982\)
\(\displaystyle a(60) = 23774520252107190517434120165783910805559526\)
\(\displaystyle a(61) = 139192917599487406399610517797537649319045588\)
\(\displaystyle a(62) = 815251949170680541251403116703129290357707337\)
\(\displaystyle a(63) = 4776728500032637956234829368622464147652032227\)
\(\displaystyle a(64) = 27998076316176086079224938446989407122125255640\)
\(\displaystyle a(65) = 164164747983407206095663586356621241756989692758\)
\(\displaystyle a(66) = 962899723510559831534730056260650871313121007082\)
\(\displaystyle a(67) = 5649724128182467751348362082000925685413487844203\)
\(\displaystyle a(68) = 33159976490667495390836325268811375840675436280805\)
\(\displaystyle a(69) = 194687437817124989476136226815661248709839422919052\)
\(\displaystyle a(70) = 1143390324738929950322931547896866997614933240427062\)
\(\displaystyle a(71) = 6717075816659932439106369307230371351837059821129959\)
\(\displaystyle a(72) = 39472222617567065893526581733286224684984881391421357\)
\(\displaystyle a(73) = 232019823272962770825863481261504684660238221245542941\)
\(\displaystyle a(74) = 1364198297214212644742677659023579795102577564022720982\)
\(\displaystyle a(75) = 8023163976560805681089290596909567938932218393608130164\)
\(\displaystyle a(76) = 47198322475460265120179812708020196792133548549296070054\)
\(\displaystyle a(77) = 277726473983378035070476877129627008896669375140180679871\)
\(\displaystyle a(78) = 1634613282893347090249460410398653518137333323437971040982\)
\(\displaystyle a(79) = 9623146607230625338888950471228549580181446113498762123721\)
\(\displaystyle a(80) = 56665787086514085393543079716837675961635122334458021509512\)
\(\displaystyle a(81) = 333752093093277083527561974959816424529823676158362597476533\)
\(\displaystyle a(82) = 1966182902088539667474721708041591861087820245756243239299100\)
\(\displaystyle a(83) = 11585595422906354701552706949110066485846709572820029741332871\)
\(\displaystyle a(84) = 68281820583920195716674908121632264852566074756212735494322826\)
\(\displaystyle a(85) = 402514896912853805225130613723153382750275276302796590058900777\)
\(\displaystyle a(86) = 2373268492146718090659006842551730443119646772551017696434767422\)
\(\displaystyle a(87) = 13995803193486585735548207021418243105651002296073171243894174573\)
\(\displaystyle a(88) = 82553002748208111686190451279827158322823540672545083286580420104\)
\(\displaystyle a(89) = 487023748175581387568847085803204554029905705223156911970286508349\)
\(\displaystyle a(90) = 2873742181913597878885820225829150720087677246774723594622964427768\)
\(\displaystyle a(91) = 16959932140768122313298595622428446526239889871153074910279394396226\)
\(\displaystyle a(92) = 100109976975920128367715335990842546787928118246285937852050867307400\)
\(\displaystyle a(93) = 591025081210527384840099194188097609331056654766749558149271207603696\)
\(\displaystyle a(94) = 3489861299859557549510267368699270904354461084742314768699135999250716\)
\(\displaystyle a(95) = 20610217976554244237729987476792416031429103551822125512260855820780851\)
\(\displaystyle a(96) = 121738422764917085919508974274460521253591615229989246890532927724119353\)
\(\displaystyle a(97) = 719187242877578868631483549721349142954758436995086133745601822072614011\)
\(\displaystyle a(98) = 4249365555301484886938556270611079217306640531085838909817128568187860490\)
\(\displaystyle a(99) = 25111500401922845557428216957747716781956832159458010957796541493372077926\)
\(\displaystyle a(100) = 148417926770149328535475382750415019391830447256059789873847717513190288890\)
\(\displaystyle a(101) = 877331868738583230522001077606678756966449447424485753531936272332338954420\)
\(\displaystyle a(102) = 5186854322723271223508213828855432676145744205844180251594540054161077838288\)
\(\displaystyle a(103) = 30669421741871654964612430862780125538082534098719557308834860438493431798525\)
\(\displaystyle a(104) = 181370788739489372218977327170762445552327308369477924817575025264537734772899\)
\(\displaystyle a(105) = 1072724430539464144514934660124135560810258476261739927287999108105958431157766\)
\(\displaystyle a(106) = 6345516358915953075780834022291794766014791397528855434508239119939778603928231\)
\(\displaystyle a(107) = 37540724797900507786461778329480436121644688859003689801058705317271765793226638\)
\(\displaystyle a(108) = 222123331601767273520301421265230700469965722309695631573858564879283477847460226\)
\(\displaystyle a(109) = 1314439268221565570017546915378422208118671513566441925570577671560087764350935731\)
\(\displaystyle a(110) = 7779303223377488510258824202128495107994674670597401907062006375914288520377091309\)
\(\displaystyle a(111) = 46046193921935680999682021800450468084084072195569284201866983385269917694982278883\)
\(\displaystyle a(112) = 272582958216851737355786745065903195484876276768831496508475481142131375110126362460\)
\(\displaystyle a(113) = 1613818433688716620061632856490910953472140358281205155170228498018197827112717663484\)
\(\displaystyle a(114) = 9555661605716007553367409380109379263768616527322823224979008942152335491695213447749\)
\(\displaystyle a(115) = 56586926020530394000789852695988523146765703598578916536762588189001779534472909084456\)
\(\displaystyle a(116) = 335135048224029388361226847991928283154172553632258578465107299073798788736068272857423\)
\(\displaystyle a(117) = 1985048747617334744991558665723554136478275336942211524087559340580521941933057218258004\)
\(\displaystyle a(118) = 11758970021178296122193649569403287687040276362756374983987477227969053078562689888774294\)
\(\displaystyle a(119) = 69664798632295935920658679812638751124219360842589463696366682706965546277629733037867567\)
\(\displaystyle a(120) = 412764864435902417075954267862044148642181073803807104621382634966847231048897612960384932\)
\(\displaystyle a(121) = 2445887887504342121343691438384615778509415888342413146776223647102353540945031884246732557\)
\(\displaystyle a(122) = 14494863604271765219011705162248488363619852588649339408782792523940445186629206550082987096\)
\(\displaystyle a(123) = 85908230452832928609510715008159360947392353419376264200038979929560825896198364833662672887\)
\(\displaystyle a(124) = 509210999403696914666814655385862115341151151032907784183774243339974781264835570473441924638\)
\(\displaystyle a(125) = 3018578443443693966415327726580900436025545946137571580261998916084420617567608904675870378079\)
\(\displaystyle a(126) = 17895679150394004657361966812042438794858784137385365000335868253030511314946599764054018787755\)
\(\displaystyle a(127) = 106104618478796948315869507107207639503912737697050974242683464330739421755794780329640653296293\)
\(\displaystyle a(128) = 629158615328619067266800298245506838037552127949738880206532365545973215526766750122968103833998\)
\(\displaystyle a(129) = 3730999152583448532129257898424074873674187131702354155256210168400735689659433816268189228931116\)
\(\displaystyle a(130) = 22127313840598029733610969633177210351659933861501901856857705305568690169014776224730411680946331\)
\(\displaystyle a(131) = 131241201505008543167038244270428442167307630986495612568890688392943068384520917818948849052875276\)
\(\displaystyle a(132) = 778482911093603030902580941006242727701463480382880540525241849753405989134190662020766327990757673\)
\(\displaystyle a(133) = 4618115530880340630161989672852606204926749887818848542231042608308049441122440553689305244122377454\)
\(\displaystyle a(134) = 27397868679216403011216747504059897484551166870042984674969957344979599482358170708566823340196215544\)
\(\displaystyle a(135) = 162556562592600393207051180917015877050285087922073650020271921973405731086365782699121398632530259276\)
\(\displaystyle a(136) = 964556011540670020778420591698688080604827638080868669454700977377046634777528628169737612697126317130\)
\(\displaystyle a(137) = 5723808594024555881688355577943984006104117440857917511552108409094233215325637998129259666555367495577\)
\(\displaystyle a(138) = 33968545949669923019805948966667226450708993322321046572089010401794476046488854038746646755179078312387\)
\(\displaystyle a(139) = 201605569426810981906070973998697391043133552408770481419678819896290046518429822025708452576273927221750\)
\(\displaystyle a(140) = 1196633971986049972566351404504304272881374505641866402397135563942770639923258946324056382822972797947657\)
\(\displaystyle a(141) = 7103181228062016919612798568402317420761875499904120997722448477868357261407768430493566030013984343922178\)
\(\displaystyle a(142) = 42167394500793660523749620713585733264941922813187330504671791789638880676744922346230725550931731365104096\)
\(\displaystyle a(143) = 250341294333247301679467545094681167194699838867502822692116146252007321742952174795028957310383843362471940\)
\(\displaystyle a(144) = 1486345023138899536741499724811092978998666803731309307123673555564876921529343252760436071279354887757617226\)
\(\displaystyle a(145) = 8825468215041887677247114745288404079858265801592261601855134814669228771549203331006248243876253179339890890\)
\(\displaystyle a(146) = 52406654463211971019382280241709251524275851259053490199238981197413870480016340063976099939254924850657754721\)
\(\displaystyle a(147) = 311218397201521795651565228930867094943463949833690676365361946665253221698795106614525876272396827327116356993\)
\(\displaystyle a(148) = 1848305799286991249258572788232269550967870118429643889160824151998880217283178048093542460556874859206888419749\)
\(\displaystyle a(149) = 10977709439738963111787947770870955073003099316345018667811063915023697360967236849334689974951478255118000746721\)
\(\displaystyle a{\left(n + 150 \right)} = \frac{278337545024 n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) a{\left(n \right)}}{12493 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{198912 \left(n + 1\right) \left(n + 2\right) \left(n + 3\right) \left(423624884 n + 1606661391\right) a{\left(n + 1 \right)}}{62465 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{224 \left(n + 2\right) \left(n + 3\right) \left(10494300021479 n^{2} + 91039345069319 n + 195162024006072\right) a{\left(n + 2 \right)}}{62465 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{32 \left(n + 3\right) \left(3758283797139653 n^{3} + 55962020907505593 n^{2} + 272662805567696938 n + 435910378983463896\right) a{\left(n + 3 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(4999 n^{2} + 1499559 n + 112436041\right) a{\left(n + 149 \right)}}{31 \left(n + 151\right) \left(n + 153\right)} - \frac{\left(463891964 n^{3} + 206896992126 n^{2} + 30756844682755 n + 1523982549316164\right) a{\left(n + 148 \right)}}{37479 \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(45404817705 n^{4} + 26837671553878 n^{3} + 5948463937646073 n^{2} + 585960326599150976 n + 21644562918435528072\right) a{\left(n + 147 \right)}}{74958 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(798939042848 n^{4} + 469122415443012 n^{3} + 103294265071806085 n^{2} + 10108111004329084125 n + 370920994124966151282\right) a{\left(n + 146 \right)}}{37479 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(107946793807077 n^{4} + 62967234765697331 n^{3} + 13773284907863680668 n^{2} + 1338951244758332883814 n + 48810153722000581694700\right) a{\left(n + 145 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(4666497158271491 n^{4} + 2704145806848176050 n^{3} + 587608329988930469413 n^{2} + 56748033166397791420586 n + 2055098196118434375209460\right) a{\left(n + 144 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{16 \left(29002772719163877 n^{4} + 664914357434694254 n^{3} + 5547732395570922191 n^{2} + 20078050062593882718 n + 26678309963653662120\right) a{\left(n + 4 \right)}}{62465 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(41457687380926186 n^{4} + 23866075046630131933 n^{3} + 5152007940894282412643 n^{2} + 494285573185876436196140 n + 17782757375623707829941168\right) a{\left(n + 143 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{32 \left(350654584515153557 n^{4} + 10521580943626839108 n^{3} + 111511990147278520117 n^{2} + 504775870093154017182 n + 832119591491875201896\right) a{\left(n + 5 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(1234281767999569311 n^{4} + 705881114943610960186 n^{3} + 151380316484163176469507 n^{2} + 14428266438468594357773876 n + 515679266174968341987846456\right) a{\left(n + 142 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{8 \left(2281258867936378659 n^{4} - 348361557477688006448 n^{3} - 8705810610447740878161 n^{2} - 67967128138293758822426 n - 174149696503922693314104\right) a{\left(n + 7 \right)}}{62465 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{8 \left(6909629267174874643 n^{4} + 309363290978995530474 n^{3} + 4332951145796567847737 n^{2} + 24700459425605174519130 n + 50084175959113867396056\right) a{\left(n + 6 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(31199860879712046023 n^{4} + 17726309569859876104894 n^{3} + 3776641528068354137013565 n^{2} + 357603094709122320184019702 n + 12697514559791972598355740528\right) a{\left(n + 141 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(84483002652246316542 n^{4} + 47686209266058199093403 n^{3} + 10093443780919453719799719 n^{2} + 949500844447547285011980988 n + 33494588141577798893804126430\right) a{\left(n + 140 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(789246605671553827430 n^{4} + 442590583229914886331121 n^{3} + 93071409000416196986477053 n^{2} + 8698440427648942149328867856 n + 304853627635313949483817529550\right) a{\left(n + 139 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{4 \left(1291605610851817764355 n^{4} + 24527651252176049039898 n^{3} + 64319824646584557776201 n^{2} - 1052304678805693222193310 n - 5266952865741453407523504\right) a{\left(n + 8 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{16 \left(4616138962103613295421 n^{4} + 145533358712344236335331 n^{3} + 1651731709170751141251637 n^{2} + 7805656683867699276138705 n + 12269021027902946302950486\right) a{\left(n + 9 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - 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\frac{\left(12821481971512922811373588381553061958092593 n^{4} + 3582175188620510095496655787621396524993343110 n^{3} + 375189289137849657846792952839130202755163568091 n^{2} + 17459696865259841192643412525527441071321109168882 n + 304593588904454120806376968896547476264011655997536\right) a{\left(n + 68 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(13574886937241115156656918638654831384650937 n^{4} + 4927039403541461363668582213747126084673061390 n^{3} + 670849902357830759794449093539535428820957670463 n^{2} + 40610460461313945110847926132749719688942052290614 n + 922220652088575273193625675705988728414197996500900\right) a{\left(n + 89 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(13903116853620682123283777691830643928021315 n^{4} + 3808069613136587044097265276199201543330292094 n^{3} + 391022017507748508283479702537946721844553263941 n^{2} + 17839884312773798390593448343841840444270726300386 n + 305137461624399437090675189392732888610048473391040\right) a{\left(n + 67 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(14187029241091476303824302457368037951347959 n^{4} + 5421607567544290378071139963614723072410856826 n^{3} + 776733819357869170617784123768127156834756249913 n^{2} + 49443540965032535896381602876789771484339130397662 n + 1179922689607339782010186635126125742739141822625208\right) a{\left(n + 93 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(15027675067930076596321919469235155753564481 n^{4} + 4857885574533149570421661436630711947242666014 n^{3} + 588624677471771309739047629207567404656234264195 n^{2} + 31684496181794905064842573625056941058838210814334 n + 639265470134833987731949148009061773019939302950272\right) a{\left(n + 82 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(16810364314681915478834077382524181547458955 n^{4} + 5114331171140164790924184731707019331824901112 n^{3} + 583408932160056543276548180109279372420735949926 n^{2} + 29574522472350108205186057922097951465129517480331 n + 562132008643835781567229709098441824123429447164382\right) a{\left(n + 75 \right)}}{187395 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(16917085982837129232622238189303774908178267 n^{4} + 4987167964296614013242625061530361235955969234 n^{3} + 551184309849795312232647617910761632254779771625 n^{2} + 27067077218560476255288787352694747408785777741170 n + 498313094540020156210903733280682764978488229144872\right) a{\left(n + 72 \right)}}{249860 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(19447638097945251543547159492447283742465983 n^{4} + 5623488631460672563612677794863343924352366396 n^{3} + 609658803556239612671450294957673582377272649167 n^{2} + 29369632417267871502890497188385475802366492882438 n + 530465129784987554240082541725012632032534554181176\right) a{\left(n + 71 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(21946296205450597311075617089885736188002989 n^{4} + 7710688819325553096239774522912716607457593422 n^{3} + 1015874633689072925420765442571457443136127459979 n^{2} + 59481836552919753129726560791870339074055328960858 n + 1305978060847157453611616444583698973888323071986600\right) a{\left(n + 87 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(24482746638338137509126643229718955036261175 n^{4} + 7586502178092736781469886856825739897557678298 n^{3} + 881389897245803244110489057373899379691749349241 n^{2} + 45501606172239208325219870744057877638800272780270 n + 880708582656921367383477400792092490426844937929912\right) a{\left(n + 76 \right)}}{249860 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(29698520474177230173394550038311861362896109 n^{4} + 9224286064522895878156220091749904447386820910 n^{3} + 1074162375667418886443564886314837856174627421903 n^{2} + 55582311894390243760822898248797268417665536632622 n + 1078328190781200704880350886228113400840748005560232\right) a{\left(n + 78 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(33386134383019085769304612000536483834624693 n^{4} + 10609883026716409301396762988472135630201845682 n^{3} + 1263651610224244998542599432384197694281978615483 n^{2} + 66851241053032870417956530260484982116845191227726 n + 1325492044808671233159124539420599628437264744240424\right) a{\left(n + 77 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(40207526063484251643015848829345964355648715 n^{4} + 13052008796592943231654533144362394103599743072 n^{3} + 1588709644589905554976105727764159064538334427901 n^{2} + 85940195041310697820938481834361296492408070629756 n + 1743202254114706152794686325116194827819476026891732\right) a{\left(n + 80 \right)}}{374790 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(40948833576325753508933723898122068090445753 n^{4} + 14128166046517884331446605369334579160930597702 n^{3} + 1828240074787000332053058243753381866164466558323 n^{2} + 105165326932668467085866548240667762543836458529766 n + 2268924565254253765997125643744775773405198584686720\right) a{\left(n + 85 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(41083859694186375341254121750939835531364533 n^{4} + 13598273172889863620352276516541130653495886018 n^{3} + 1687469552592244789798335246819497221519282141203 n^{2} + 93050027156384087043735914569130939798281119939166 n + 1923718879662029947610318899288744663587832010390544\right) a{\left(n + 81 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} + \frac{\left(55919575598076246126309263748650325424574741 n^{4} + 18706083153296563797812890805665321032391950038 n^{3} + 2346695042395156594010276296680139848483013363435 n^{2} + 130849869269268814659816992501449073818789479297106 n + 2736203680990659428921289912625720899516147731895768\right) a{\left(n + 83 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)} - \frac{\left(63484090352259739492766929774116961116750815 n^{4} + 21573046581154930118526490125490939899089645454 n^{3} + 2749344546724608221178335372742497828739825362941 n^{2} + 155742106067338205677583969235571908884837631036822 n + 3308702706487571360859247900372167039180329051207304\right) a{\left(n + 84 \right)}}{749580 \left(n + 149\right) \left(n + 150\right) \left(n + 151\right) \left(n + 153\right)}, \quad n \geq 150\)