Av(13452, 13542, 31452, 31542, 34152)
Counting Sequence
1, 1, 2, 6, 24, 115, 614, 3504, 20885, 128292, 805653, 5145429, 33303599, 217908781, 1438724862, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{4} \left(x^{2}-3 x +1\right) F \left(x
\right)^{5}-x^{3} \left(9 x^{2}-19 x +6\right) F \left(x
\right)^{4}+\left(2 x^{5}+8 x^{4}-27 x^{3}+5 x^{2}+4 x -1\right) F \left(x
\right)^{3}+\left(-15 x^{4}+20 x^{3}+23 x^{2}-21 x +4\right) F \left(x
\right)^{2}+\left(x^{4}+11 x^{3}-43 x^{2}+28 x -5\right) F \! \left(x \right)-\left(3 x -2\right) \left(2 x^{2}-4 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 115\)
\(\displaystyle a(6) = 614\)
\(\displaystyle a(7) = 3504\)
\(\displaystyle a(8) = 20885\)
\(\displaystyle a(9) = 128292\)
\(\displaystyle a(10) = 805653\)
\(\displaystyle a(11) = 5145429\)
\(\displaystyle a(12) = 33303599\)
\(\displaystyle a(13) = 217908781\)
\(\displaystyle a(14) = 1438724862\)
\(\displaystyle a(15) = 9571907890\)
\(\displaystyle a(16) = 64101679227\)
\(\displaystyle a(17) = 431736779901\)
\(\displaystyle a(18) = 2922449260601\)
\(\displaystyle a(19) = 19870384374339\)
\(\displaystyle a(20) = 135641064945713\)
\(\displaystyle a(21) = 929240910500579\)
\(\displaystyle a(22) = 6386603302226684\)
\(\displaystyle a(23) = 44023956710482330\)
\(\displaystyle a(24) = 304281593265160497\)
\(\displaystyle a(25) = 2108306372658500908\)
\(\displaystyle a(26) = 14641271983684948936\)
\(\displaystyle a(27) = 101891127272240583862\)
\(\displaystyle a(28) = 710460827935633351673\)
\(\displaystyle a(29) = 4962847842292055280159\)
\(\displaystyle a(30) = 34726081323945323257852\)
\(\displaystyle a(31) = 243369870581272537634828\)
\(\displaystyle a(32) = 1708129797304985717387885\)
\(\displaystyle a(33) = 12005459038609995833694929\)
\(\displaystyle a(34) = 84489911897605221699486500\)
\(\displaystyle a(35) = 595342075793252768288493694\)
\(\displaystyle a(36) = 4199853386788803485520836431\)
\(\displaystyle a(37) = 29660615540345068794061072896\)
\(\displaystyle a(38) = 209690856811649720554969284270\)
\(\displaystyle a(39) = 1483914625019128791358683978884\)
\(\displaystyle a(40) = 10511070724524819258146599088440\)
\(\displaystyle a(41) = 74520153422794373693400942243514\)
\(\displaystyle a(42) = 528774868262578531523892021124834\)
\(\displaystyle a(43) = 3755095545889000304616460444568088\)
\(\displaystyle a(44) = 26687525673866155672433935803770268\)
\(\displaystyle a(45) = 189809454428188501933013280325376147\)
\(\displaystyle a(46) = 1350938686994473589798643322731872894\)
\(\displaystyle a(47) = 9621629950820172901656730176426908846\)
\(\displaystyle a(48) = 68571655760634842582733173114079328694\)
\(\displaystyle a(49) = 489003659657078160600474562096267970101\)
\(\displaystyle a(50) = 3489315739574008146588344423833012551087\)
\(\displaystyle a(51) = 24912594062930806323511796803719859487893\)
\(\displaystyle a(52) = 177966629698290257265458493872775542455187\)
\(\displaystyle a(53) = 1272008766810252555790503039331313768437318\)
\(\displaystyle a(54) = 9096304711031514816770277318433069566641166\)
\(\displaystyle a(55) = 65081145716526913705034507648326183604752304\)
\(\displaystyle a(56) = 465857436368946284747851516994186600083314050\)
\(\displaystyle a(57) = 3336193926038315050454263318322033727309519351\)
\(\displaystyle a(58) = 23902486822246007998320346662763857478462487188\)
\(\displaystyle a(59) = 171325462747608776816169032121733326882121288682\)
\(\displaystyle a(60) = 1228518133256716043582894542727952249646937578065\)
\(\displaystyle a(61) = 8812845763567082133218384746467549572478332938255\)
\(\displaystyle a(62) = 63244112334153007931068303247942859445189892382720\)
\(\displaystyle a(63) = 454033634052676934103664973001217568453670200728740\)
\(\displaystyle a(64) = 3260730089321017783053273761179414710940612995467863\)
\(\displaystyle a(65) = 23425869479590318337431680140034863394589310832297765\)
\(\displaystyle a(66) = 168354989638325608101167693314130120875303263506253804\)
\(\displaystyle a(67) = 1210322868730434220667177372118082671360315727245246340\)
\(\displaystyle a(68) = 8703966748602328373022471175384596308805668938690241220\)
\(\displaystyle a{\left(n + 69 \right)} = - \frac{2145 \left(n + 1\right) \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{16 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(n + 2\right) \left(2 n + 3\right) \left(702184 n^{2} + 4018079 n + 5773516\right) a{\left(n + 1 \right)}}{64 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(235 n^{2} + 31516 n + 1056496\right) a{\left(n + 68 \right)}}{2 \left(n + 67\right) \left(n + 71\right)} - \frac{\left(153413 n^{3} + 30726732 n^{2} + 2051154163 n + 45636029460\right) a{\left(n + 67 \right)}}{24 \left(n + 67\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(40402187 n^{4} + 10729982198 n^{3} + 1068504707581 n^{2} + 47285000606938 n + 784609684430016\right) a{\left(n + 66 \right)}}{192 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(147639598 n^{4} + 1823656801 n^{3} + 8489034741 n^{2} + 17611864790 n + 13701015230\right) a{\left(n + 2 \right)}}{128 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1737288148 n^{4} + 455387357667 n^{3} + 44759140396994 n^{2} + 1955061269811405 n + 32020688007132954\right) a{\left(n + 65 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{25 \left(2017458344 n^{4} + 27000786667 n^{3} + 137132414488 n^{2} + 312830921003 n + 269763700056\right) a{\left(n + 3 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(23235886495 n^{4} + 6029880561167 n^{3} + 586745940934688 n^{2} + 25372841278736920 n + 411413935531691376\right) a{\left(n + 64 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(206863224248 n^{4} + 54992295822609 n^{3} + 5472098351940862 n^{2} + 241590799540141443 n + 3993407830929459126\right) a{\left(n + 63 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(294368632728 n^{4} + 3473927124319 n^{3} + 9872532444945 n^{2} - 14981839877500 n - 71468866590666\right) a{\left(n + 4 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7009233775961 n^{4} + 1696784831262837 n^{3} + 153827794085358334 n^{2} + 6189362013836940990 n + 93247058079870094764\right) a{\left(n + 62 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(10113241866724 n^{4} + 338198236831735 n^{3} + 3702094015023950 n^{2} + 16859686099859675 n + 27761444232335946\right) a{\left(n + 5 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(183194001991640 n^{4} - 73737878880446 n^{3} - 51166578570920369 n^{2} - 453969721632817321 n - 1136165615024160894\right) a{\left(n + 6 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(218281568683475 n^{4} + 52743736012696770 n^{3} + 4776805995524500777 n^{2} + 192176032859293427112 n + 2897778994758860030982\right) a{\left(n + 61 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7402619901340187 n^{4} + 1765011447347501652 n^{3} + 157752115783553661817 n^{2} + 6264022113223606067256 n + 93238287258855218336772\right) a{\left(n + 60 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(22191135638068315 n^{4} + 709322089263594798 n^{3} + 8428025943886686461 n^{2} + 44099759260066003206 n + 85695309961552010220\right) a{\left(n + 7 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(91086851275342523 n^{4} + 21372526156688262478 n^{3} + 1879880240350856484919 n^{2} + 73462209636814753699148 n + 1076143713700403798333856\right) a{\left(n + 59 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(436021605335092969 n^{4} + 100508951608476694437 n^{3} + 8684777917886067646475 n^{2} + 333391482428163369449313 n + 4797381987945343479598080\right) a{\left(n + 58 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2536573638916258405 n^{4} + 85957187843108333564 n^{3} + 1089549642800537231759 n^{2} + 6119413427027976284836 n + 12842249964648870648936\right) a{\left(n + 8 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3297773237259923850 n^{4} + 745303932617683742653 n^{3} + 63131736449977288562571 n^{2} + 2375445868049650632661790 n + 33499199250123676293923892\right) a{\left(n + 57 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(42961475477616931681 n^{4} + 1579751887071302575347 n^{3} + 21708815430061853651672 n^{2} + 132063831871488179181516 n + 299901212487664699335204\right) a{\left(n + 9 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(76808308241162490495 n^{4} + 16953174586251190235578 n^{3} + 1401952096572971109836325 n^{2} + 51478198556222870645091650 n + 708135552306043077943090968\right) a{\left(n + 56 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(91749947722823911371 n^{4} + 14190648596863478647338 n^{3} + 675441576831345042441381 n^{2} + 6582776975828857032150702 n - 159758798404967841515864744\right) a{\left(n + 54 \right)}}{1024 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(152218632569691298115 n^{4} + 32394201545327128853832 n^{3} + 2578532550279009457177921 n^{2} + 90961165397446036583550264 n + 1199493508504786268652583884\right) a{\left(n + 55 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(429313581062378521879 n^{4} + 17325725431091481623108 n^{3} + 261920598170244513064700 n^{2} + 1758141435986343392898949 n + 4422123743571418524938784\right) a{\left(n + 10 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1971059264981038690359 n^{4} + 446426083033959255727855 n^{3} + 37853327346224445331390767 n^{2} + 1424306180392529027160116003 n + 20068240053619060452317279760\right) a{\left(n + 53 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3931832256741707354551 n^{4} + 177716889485307428421708 n^{3} + 3021890282173536063177962 n^{2} + 22922211342481308034900815 n + 65478107519915821178554764\right) a{\left(n + 11 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(42385912053065919534199 n^{4} + 9152326979000713434379432 n^{3} + 741223302992689579938137855 n^{2} + 26684362138407435502633824842 n + 360300637249061738797934958096\right) a{\left(n + 52 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(76272860206548031541173 n^{4} + 3844864924009331989185946 n^{3} + 72967973020603793294476643 n^{2} + 617998087682980338949652210 n + 1971113073523156229727300468\right) a{\left(n + 12 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(119799252806196705045758 n^{4} + 6617134170988312659251656 n^{3} + 137468704129416950899678541 n^{2} + 1273036521223900772166660613 n + 4433830435464954066893467042\right) a{\left(n + 13 \right)}}{256 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(138668549985878719425449 n^{4} + 29138976566398334202079068 n^{3} + 2297344444000507054456800457 n^{2} + 80541087316298781178831950288 n + 1059398952063320394836714213274\right) a{\left(n + 51 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1346244465440653430663463 n^{4} + 276519885050316744899319914 n^{3} + 21315720693393827204346233127 n^{2} + 730855043580593155233567508228 n + 9404439156471865731631516936680\right) a{\left(n + 50 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1865706855020779402988606 n^{4} + 337619766548255929754887401 n^{3} + 21916814847860438564820349849 n^{2} + 595062656304366902736826663278 n + 5500972362511773333660589940712\right) a{\left(n + 47 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2445116652422354506802171 n^{4} + 491790316042958704462705982 n^{3} + 37138827745920936521504976115 n^{2} + 1248049798566496653796558550536 n + 15747221141753126004219370652640\right) a{\left(n + 49 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2993753853403555478783536 n^{4} + 178795439951801145968923292 n^{3} + 4012572965576814262975565705 n^{2} + 40103530404171344435625603937 n + 150599267919843110847999961158\right) a{\left(n + 14 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(21068231881131528479674309 n^{4} + 1348135379299291062799367623 n^{3} + 32396802315192558565719149297 n^{2} + 346498340861416037242538295743 n + 1391622802111359221531063027316\right) a{\left(n + 15 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(22658408003102384695413369 n^{4} + 4471916314078392263455920242 n^{3} + 331832380923228756606123476655 n^{2} + 10971972617920553668403354537326 n + 136392186774435429109246503322296\right) a{\left(n + 48 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(123569104538644310114470026 n^{4} + 8416101533075995817586444593 n^{3} + 215185680204527765972857584561 n^{2} + 2447869308255199158750396871750 n + 10452784915940647484121485217510\right) a{\left(n + 16 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(426368762686738385856868043 n^{4} + 79962053782487559764511955834 n^{3} + 5610078521066060508405563446909 n^{2} + 174497166493236758823508010615318 n + 2030063323590418217385920210630784\right) a{\left(n + 46 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(709763180751513107925949002 n^{4} + 130521848228277050490392575061 n^{3} + 8989616094463547913404417028567 n^{2} + 274826730727480146733939644890104 n + 3146545883550275627256575964126210\right) a{\left(n + 45 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1213142275404096173535906245 n^{4} + 87526676820894000217643499232 n^{3} + 2370150458974274731882004258791 n^{2} + 28549128749256482272534517297024 n + 129060101557903801655155119499920\right) a{\left(n + 17 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1641786441114063188318987711 n^{4} + 295582175485002365766743047010 n^{3} + 19938935012288379294329714712583 n^{2} + 597263189039959938322631050369336 n + 6703079484537241412642815439511297\right) a{\left(n + 44 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(5051085842012942130708344447 n^{4} + 384736565025205268053474412448 n^{3} + 10997530534058925618564113146861 n^{2} + 139815992257968946511708707625052 n + 667039952062777219122553503645420\right) a{\left(n + 18 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(12344385918221301341630451622 n^{4} + 2174294477931303696994658342770 n^{3} + 143521873227747080531774449661567 n^{2} + 4207745421425345651510894963773097 n + 46229632279516446561208378869002496\right) a{\left(n + 43 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(18145364715482262435296643901 n^{4} + 1455505979224421070501951977408 n^{3} + 43810995353101779401993538063335 n^{2} + 586477137597191142602537434591624 n + 2945930762900589930864504722129772\right) a{\left(n + 19 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(19073163246089045457505447835 n^{4} + 1608009850211436128793641544668 n^{3} + 50868739292604990669238579190083 n^{2} + 715628613054641261361870861582122 n + 3777494689705630505405758672225892\right) a{\left(n + 20 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(79477940597878402843018610261 n^{4} + 13688083847262389209385568137072 n^{3} + 883575481330240066707419889413683 n^{2} + 25335648111729962276944260767226700 n + 272281138248748951425933934291723788\right) a{\left(n + 42 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(160793411296706179038204237099 n^{4} + 14222881860468674413064202365404 n^{3} + 472037463901508561915331401618463 n^{2} + 6966469444609884182605407340743242 n + 38574590799811433598159726020865840\right) a{\left(n + 21 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(224501774614696983138176899735 n^{4} + 37786307042831947076495909823758 n^{3} + 2383914578623164744910525310686649 n^{2} + 66814696458247534467322801837611226 n + 701922836420470335439027649407873788\right) a{\left(n + 41 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(468109857861216042447835020884 n^{4} + 45329510977577195286365832659110 n^{3} + 1646677346298972645268303017243583 n^{2} + 26595505822747774547527751925679217 n + 161133460941834019776338107882405332\right) a{\left(n + 23 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(563988132692675320070057095021 n^{4} + 92719561424331622999309619891786 n^{3} + 5714009004689790820575385885593593 n^{2} + 156445435131672996163407636366974728 n + 1605643459925282062618422373485556260\right) a{\left(n + 40 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(814511191584175713394865290077 n^{4} + 75455669155327180869380726928698 n^{3} + 2622539999086509128251392003222915 n^{2} + 40528964752403857208176807546483918 n + 234977891025247448115079045935867384\right) a{\left(n + 22 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1270438242868496944349248752703 n^{4} + 203892293485622584685653035762990 n^{3} + 12266946968476764965229707421598049 n^{2} + 327903039326548960384502571888116274 n + 3285792298916521849226818965938737824\right) a{\left(n + 39 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1289800930694240884846804344209 n^{4} + 201958915933780294727320648725862 n^{3} + 11855172231669781206149856250112878 n^{2} + 309202142536102159985538711983095715 n + 3023290014942033819053172823256178006\right) a{\left(n + 38 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3402640943970975497549036173923 n^{4} + 385555204028835431861658800199518 n^{3} + 16383140264649935630477470661481341 n^{2} + 309407530310646320020222714463550082 n + 2191282993015071755525484737032145600\right) a{\left(n + 27 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3734994513176976837276237801845 n^{4} + 392766533725825581537902205251356 n^{3} + 15491248508710996034368868980739449 n^{2} + 271597683532428578663848817645788502 n + 1785917182541769787235328760548711108\right) a{\left(n + 25 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(3918906120140980926086415671449 n^{4} + 395867848982142605097208346494762 n^{3} + 14999801843459566246108344690733607 n^{2} + 252667670344546497011605589433329222 n + 1596427556260048464432248411694546264\right) a{\left(n + 24 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(3944012103520209303656419233028 n^{4} + 586744535212034533499156852189471 n^{3} + 32725672925805441044881377108522074 n^{2} + 811040856803020551139033046141304307 n + 7535724535547288834342448419118101592\right) a{\left(n + 36 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3974169156407476918709795945657 n^{4} + 575706214060082561397617074337284 n^{3} + 31267549962569425856578974152283531 n^{2} + 754590322553815116288148977901799172 n + 6827579385663410890567351931477081656\right) a{\left(n + 35 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(4737677638289416679609595343639 n^{4} + 723323033488004628459167356203148 n^{3} + 41401561407640373198093120710987865 n^{2} + 1052941935996603425438145037552967936 n + 10039420770762418702116379209933698484\right) a{\left(n + 37 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(6477528922882912454602034951163 n^{4} + 707716099098347993196622776029840 n^{3} + 28998741186452751910194301457314317 n^{2} + 528142681527488848118232697442650156 n + 3607321251397618916991065859636119880\right) a{\left(n + 26 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7301676348292086109783244707997 n^{4} + 856663703648890593982996219573718 n^{3} + 37689007439533836438217876937570386 n^{2} + 736918881709161972639641978036471845 n + 5403026605242558270299664650739738612\right) a{\left(n + 28 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(8185913756073408436673452250954 n^{4} + 1153837941022339269851644855809626 n^{3} + 60977495104996851836106247879804627 n^{2} + 1431949282834147287842232675094288099 n + 12607610838737263039674935774691407598\right) a{\left(n + 34 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(9478096638003299160009932781277 n^{4} + 1149746230092757574052210483332420 n^{3} + 52297548122479333106091332009680949 n^{2} + 1057168133622924071463386008859046136 n + 8013141247760999226696287731400912514\right) a{\left(n + 29 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(11933911136070629195402527182599 n^{4} + 1541867740313376804304781300837652 n^{3} + 74693850064294164013869533490733426 n^{2} + 1607984502759780705544117957140858435 n + 12979312748452685168830805540310434580\right) a{\left(n + 31 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(20430733537045546383566237554539 n^{4} + 2799886232139713160089368306775546 n^{3} + 143864049856553224602272122787696145 n^{2} + 3284776634229296888020743180445967326 n + 28119972833901870703643271321371373804\right) a{\left(n + 33 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(44653728311574386136752137167319 n^{4} + 5593422808232503654088888712253922 n^{3} + 262713323334424520517769797594827201 n^{2} + 5483484044883268392775177044148525798 n + 42915509512304167190972929140634305144\right) a{\left(n + 30 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(46337294869700488078397754611923 n^{4} + 6168701371664858309379022915371518 n^{3} + 307907892955331938382140576742105933 n^{2} + 6829643069319863263530011844207377914 n + 56798785261461265829554496987613230856\right) a{\left(n + 32 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)}, \quad n \geq 69\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 115\)
\(\displaystyle a(6) = 614\)
\(\displaystyle a(7) = 3504\)
\(\displaystyle a(8) = 20885\)
\(\displaystyle a(9) = 128292\)
\(\displaystyle a(10) = 805653\)
\(\displaystyle a(11) = 5145429\)
\(\displaystyle a(12) = 33303599\)
\(\displaystyle a(13) = 217908781\)
\(\displaystyle a(14) = 1438724862\)
\(\displaystyle a(15) = 9571907890\)
\(\displaystyle a(16) = 64101679227\)
\(\displaystyle a(17) = 431736779901\)
\(\displaystyle a(18) = 2922449260601\)
\(\displaystyle a(19) = 19870384374339\)
\(\displaystyle a(20) = 135641064945713\)
\(\displaystyle a(21) = 929240910500579\)
\(\displaystyle a(22) = 6386603302226684\)
\(\displaystyle a(23) = 44023956710482330\)
\(\displaystyle a(24) = 304281593265160497\)
\(\displaystyle a(25) = 2108306372658500908\)
\(\displaystyle a(26) = 14641271983684948936\)
\(\displaystyle a(27) = 101891127272240583862\)
\(\displaystyle a(28) = 710460827935633351673\)
\(\displaystyle a(29) = 4962847842292055280159\)
\(\displaystyle a(30) = 34726081323945323257852\)
\(\displaystyle a(31) = 243369870581272537634828\)
\(\displaystyle a(32) = 1708129797304985717387885\)
\(\displaystyle a(33) = 12005459038609995833694929\)
\(\displaystyle a(34) = 84489911897605221699486500\)
\(\displaystyle a(35) = 595342075793252768288493694\)
\(\displaystyle a(36) = 4199853386788803485520836431\)
\(\displaystyle a(37) = 29660615540345068794061072896\)
\(\displaystyle a(38) = 209690856811649720554969284270\)
\(\displaystyle a(39) = 1483914625019128791358683978884\)
\(\displaystyle a(40) = 10511070724524819258146599088440\)
\(\displaystyle a(41) = 74520153422794373693400942243514\)
\(\displaystyle a(42) = 528774868262578531523892021124834\)
\(\displaystyle a(43) = 3755095545889000304616460444568088\)
\(\displaystyle a(44) = 26687525673866155672433935803770268\)
\(\displaystyle a(45) = 189809454428188501933013280325376147\)
\(\displaystyle a(46) = 1350938686994473589798643322731872894\)
\(\displaystyle a(47) = 9621629950820172901656730176426908846\)
\(\displaystyle a(48) = 68571655760634842582733173114079328694\)
\(\displaystyle a(49) = 489003659657078160600474562096267970101\)
\(\displaystyle a(50) = 3489315739574008146588344423833012551087\)
\(\displaystyle a(51) = 24912594062930806323511796803719859487893\)
\(\displaystyle a(52) = 177966629698290257265458493872775542455187\)
\(\displaystyle a(53) = 1272008766810252555790503039331313768437318\)
\(\displaystyle a(54) = 9096304711031514816770277318433069566641166\)
\(\displaystyle a(55) = 65081145716526913705034507648326183604752304\)
\(\displaystyle a(56) = 465857436368946284747851516994186600083314050\)
\(\displaystyle a(57) = 3336193926038315050454263318322033727309519351\)
\(\displaystyle a(58) = 23902486822246007998320346662763857478462487188\)
\(\displaystyle a(59) = 171325462747608776816169032121733326882121288682\)
\(\displaystyle a(60) = 1228518133256716043582894542727952249646937578065\)
\(\displaystyle a(61) = 8812845763567082133218384746467549572478332938255\)
\(\displaystyle a(62) = 63244112334153007931068303247942859445189892382720\)
\(\displaystyle a(63) = 454033634052676934103664973001217568453670200728740\)
\(\displaystyle a(64) = 3260730089321017783053273761179414710940612995467863\)
\(\displaystyle a(65) = 23425869479590318337431680140034863394589310832297765\)
\(\displaystyle a(66) = 168354989638325608101167693314130120875303263506253804\)
\(\displaystyle a(67) = 1210322868730434220667177372118082671360315727245246340\)
\(\displaystyle a(68) = 8703966748602328373022471175384596308805668938690241220\)
\(\displaystyle a{\left(n + 69 \right)} = - \frac{2145 \left(n + 1\right) \left(n + 2\right) \left(2 n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{16 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(n + 2\right) \left(2 n + 3\right) \left(702184 n^{2} + 4018079 n + 5773516\right) a{\left(n + 1 \right)}}{64 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(235 n^{2} + 31516 n + 1056496\right) a{\left(n + 68 \right)}}{2 \left(n + 67\right) \left(n + 71\right)} - \frac{\left(153413 n^{3} + 30726732 n^{2} + 2051154163 n + 45636029460\right) a{\left(n + 67 \right)}}{24 \left(n + 67\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(40402187 n^{4} + 10729982198 n^{3} + 1068504707581 n^{2} + 47285000606938 n + 784609684430016\right) a{\left(n + 66 \right)}}{192 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(147639598 n^{4} + 1823656801 n^{3} + 8489034741 n^{2} + 17611864790 n + 13701015230\right) a{\left(n + 2 \right)}}{128 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1737288148 n^{4} + 455387357667 n^{3} + 44759140396994 n^{2} + 1955061269811405 n + 32020688007132954\right) a{\left(n + 65 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{25 \left(2017458344 n^{4} + 27000786667 n^{3} + 137132414488 n^{2} + 312830921003 n + 269763700056\right) a{\left(n + 3 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(23235886495 n^{4} + 6029880561167 n^{3} + 586745940934688 n^{2} + 25372841278736920 n + 411413935531691376\right) a{\left(n + 64 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(206863224248 n^{4} + 54992295822609 n^{3} + 5472098351940862 n^{2} + 241590799540141443 n + 3993407830929459126\right) a{\left(n + 63 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{5 \left(294368632728 n^{4} + 3473927124319 n^{3} + 9872532444945 n^{2} - 14981839877500 n - 71468866590666\right) a{\left(n + 4 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7009233775961 n^{4} + 1696784831262837 n^{3} + 153827794085358334 n^{2} + 6189362013836940990 n + 93247058079870094764\right) a{\left(n + 62 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(10113241866724 n^{4} + 338198236831735 n^{3} + 3702094015023950 n^{2} + 16859686099859675 n + 27761444232335946\right) a{\left(n + 5 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(183194001991640 n^{4} - 73737878880446 n^{3} - 51166578570920369 n^{2} - 453969721632817321 n - 1136165615024160894\right) a{\left(n + 6 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(218281568683475 n^{4} + 52743736012696770 n^{3} + 4776805995524500777 n^{2} + 192176032859293427112 n + 2897778994758860030982\right) a{\left(n + 61 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7402619901340187 n^{4} + 1765011447347501652 n^{3} + 157752115783553661817 n^{2} + 6264022113223606067256 n + 93238287258855218336772\right) a{\left(n + 60 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(22191135638068315 n^{4} + 709322089263594798 n^{3} + 8428025943886686461 n^{2} + 44099759260066003206 n + 85695309961552010220\right) a{\left(n + 7 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(91086851275342523 n^{4} + 21372526156688262478 n^{3} + 1879880240350856484919 n^{2} + 73462209636814753699148 n + 1076143713700403798333856\right) a{\left(n + 59 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(436021605335092969 n^{4} + 100508951608476694437 n^{3} + 8684777917886067646475 n^{2} + 333391482428163369449313 n + 4797381987945343479598080\right) a{\left(n + 58 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2536573638916258405 n^{4} + 85957187843108333564 n^{3} + 1089549642800537231759 n^{2} + 6119413427027976284836 n + 12842249964648870648936\right) a{\left(n + 8 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3297773237259923850 n^{4} + 745303932617683742653 n^{3} + 63131736449977288562571 n^{2} + 2375445868049650632661790 n + 33499199250123676293923892\right) a{\left(n + 57 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(42961475477616931681 n^{4} + 1579751887071302575347 n^{3} + 21708815430061853651672 n^{2} + 132063831871488179181516 n + 299901212487664699335204\right) a{\left(n + 9 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(76808308241162490495 n^{4} + 16953174586251190235578 n^{3} + 1401952096572971109836325 n^{2} + 51478198556222870645091650 n + 708135552306043077943090968\right) a{\left(n + 56 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(91749947722823911371 n^{4} + 14190648596863478647338 n^{3} + 675441576831345042441381 n^{2} + 6582776975828857032150702 n - 159758798404967841515864744\right) a{\left(n + 54 \right)}}{1024 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(152218632569691298115 n^{4} + 32394201545327128853832 n^{3} + 2578532550279009457177921 n^{2} + 90961165397446036583550264 n + 1199493508504786268652583884\right) a{\left(n + 55 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(429313581062378521879 n^{4} + 17325725431091481623108 n^{3} + 261920598170244513064700 n^{2} + 1758141435986343392898949 n + 4422123743571418524938784\right) a{\left(n + 10 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1971059264981038690359 n^{4} + 446426083033959255727855 n^{3} + 37853327346224445331390767 n^{2} + 1424306180392529027160116003 n + 20068240053619060452317279760\right) a{\left(n + 53 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3931832256741707354551 n^{4} + 177716889485307428421708 n^{3} + 3021890282173536063177962 n^{2} + 22922211342481308034900815 n + 65478107519915821178554764\right) a{\left(n + 11 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(42385912053065919534199 n^{4} + 9152326979000713434379432 n^{3} + 741223302992689579938137855 n^{2} + 26684362138407435502633824842 n + 360300637249061738797934958096\right) a{\left(n + 52 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(76272860206548031541173 n^{4} + 3844864924009331989185946 n^{3} + 72967973020603793294476643 n^{2} + 617998087682980338949652210 n + 1971113073523156229727300468\right) a{\left(n + 12 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(119799252806196705045758 n^{4} + 6617134170988312659251656 n^{3} + 137468704129416950899678541 n^{2} + 1273036521223900772166660613 n + 4433830435464954066893467042\right) a{\left(n + 13 \right)}}{256 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(138668549985878719425449 n^{4} + 29138976566398334202079068 n^{3} + 2297344444000507054456800457 n^{2} + 80541087316298781178831950288 n + 1059398952063320394836714213274\right) a{\left(n + 51 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1346244465440653430663463 n^{4} + 276519885050316744899319914 n^{3} + 21315720693393827204346233127 n^{2} + 730855043580593155233567508228 n + 9404439156471865731631516936680\right) a{\left(n + 50 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1865706855020779402988606 n^{4} + 337619766548255929754887401 n^{3} + 21916814847860438564820349849 n^{2} + 595062656304366902736826663278 n + 5500972362511773333660589940712\right) a{\left(n + 47 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2445116652422354506802171 n^{4} + 491790316042958704462705982 n^{3} + 37138827745920936521504976115 n^{2} + 1248049798566496653796558550536 n + 15747221141753126004219370652640\right) a{\left(n + 49 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(2993753853403555478783536 n^{4} + 178795439951801145968923292 n^{3} + 4012572965576814262975565705 n^{2} + 40103530404171344435625603937 n + 150599267919843110847999961158\right) a{\left(n + 14 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(21068231881131528479674309 n^{4} + 1348135379299291062799367623 n^{3} + 32396802315192558565719149297 n^{2} + 346498340861416037242538295743 n + 1391622802111359221531063027316\right) a{\left(n + 15 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(22658408003102384695413369 n^{4} + 4471916314078392263455920242 n^{3} + 331832380923228756606123476655 n^{2} + 10971972617920553668403354537326 n + 136392186774435429109246503322296\right) a{\left(n + 48 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(123569104538644310114470026 n^{4} + 8416101533075995817586444593 n^{3} + 215185680204527765972857584561 n^{2} + 2447869308255199158750396871750 n + 10452784915940647484121485217510\right) a{\left(n + 16 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(426368762686738385856868043 n^{4} + 79962053782487559764511955834 n^{3} + 5610078521066060508405563446909 n^{2} + 174497166493236758823508010615318 n + 2030063323590418217385920210630784\right) a{\left(n + 46 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(709763180751513107925949002 n^{4} + 130521848228277050490392575061 n^{3} + 8989616094463547913404417028567 n^{2} + 274826730727480146733939644890104 n + 3146545883550275627256575964126210\right) a{\left(n + 45 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1213142275404096173535906245 n^{4} + 87526676820894000217643499232 n^{3} + 2370150458974274731882004258791 n^{2} + 28549128749256482272534517297024 n + 129060101557903801655155119499920\right) a{\left(n + 17 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1641786441114063188318987711 n^{4} + 295582175485002365766743047010 n^{3} + 19938935012288379294329714712583 n^{2} + 597263189039959938322631050369336 n + 6703079484537241412642815439511297\right) a{\left(n + 44 \right)}}{384 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(5051085842012942130708344447 n^{4} + 384736565025205268053474412448 n^{3} + 10997530534058925618564113146861 n^{2} + 139815992257968946511708707625052 n + 667039952062777219122553503645420\right) a{\left(n + 18 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(12344385918221301341630451622 n^{4} + 2174294477931303696994658342770 n^{3} + 143521873227747080531774449661567 n^{2} + 4207745421425345651510894963773097 n + 46229632279516446561208378869002496\right) a{\left(n + 43 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(18145364715482262435296643901 n^{4} + 1455505979224421070501951977408 n^{3} + 43810995353101779401993538063335 n^{2} + 586477137597191142602537434591624 n + 2945930762900589930864504722129772\right) a{\left(n + 19 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(19073163246089045457505447835 n^{4} + 1608009850211436128793641544668 n^{3} + 50868739292604990669238579190083 n^{2} + 715628613054641261361870861582122 n + 3777494689705630505405758672225892\right) a{\left(n + 20 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(79477940597878402843018610261 n^{4} + 13688083847262389209385568137072 n^{3} + 883575481330240066707419889413683 n^{2} + 25335648111729962276944260767226700 n + 272281138248748951425933934291723788\right) a{\left(n + 42 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(160793411296706179038204237099 n^{4} + 14222881860468674413064202365404 n^{3} + 472037463901508561915331401618463 n^{2} + 6966469444609884182605407340743242 n + 38574590799811433598159726020865840\right) a{\left(n + 21 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(224501774614696983138176899735 n^{4} + 37786307042831947076495909823758 n^{3} + 2383914578623164744910525310686649 n^{2} + 66814696458247534467322801837611226 n + 701922836420470335439027649407873788\right) a{\left(n + 41 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(468109857861216042447835020884 n^{4} + 45329510977577195286365832659110 n^{3} + 1646677346298972645268303017243583 n^{2} + 26595505822747774547527751925679217 n + 161133460941834019776338107882405332\right) a{\left(n + 23 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(563988132692675320070057095021 n^{4} + 92719561424331622999309619891786 n^{3} + 5714009004689790820575385885593593 n^{2} + 156445435131672996163407636366974728 n + 1605643459925282062618422373485556260\right) a{\left(n + 40 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(814511191584175713394865290077 n^{4} + 75455669155327180869380726928698 n^{3} + 2622539999086509128251392003222915 n^{2} + 40528964752403857208176807546483918 n + 234977891025247448115079045935867384\right) a{\left(n + 22 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(1270438242868496944349248752703 n^{4} + 203892293485622584685653035762990 n^{3} + 12266946968476764965229707421598049 n^{2} + 327903039326548960384502571888116274 n + 3285792298916521849226818965938737824\right) a{\left(n + 39 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(1289800930694240884846804344209 n^{4} + 201958915933780294727320648725862 n^{3} + 11855172231669781206149856250112878 n^{2} + 309202142536102159985538711983095715 n + 3023290014942033819053172823256178006\right) a{\left(n + 38 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3402640943970975497549036173923 n^{4} + 385555204028835431861658800199518 n^{3} + 16383140264649935630477470661481341 n^{2} + 309407530310646320020222714463550082 n + 2191282993015071755525484737032145600\right) a{\left(n + 27 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3734994513176976837276237801845 n^{4} + 392766533725825581537902205251356 n^{3} + 15491248508710996034368868980739449 n^{2} + 271597683532428578663848817645788502 n + 1785917182541769787235328760548711108\right) a{\left(n + 25 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(3918906120140980926086415671449 n^{4} + 395867848982142605097208346494762 n^{3} + 14999801843459566246108344690733607 n^{2} + 252667670344546497011605589433329222 n + 1596427556260048464432248411694546264\right) a{\left(n + 24 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(3944012103520209303656419233028 n^{4} + 586744535212034533499156852189471 n^{3} + 32725672925805441044881377108522074 n^{2} + 811040856803020551139033046141304307 n + 7535724535547288834342448419118101592\right) a{\left(n + 36 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(3974169156407476918709795945657 n^{4} + 575706214060082561397617074337284 n^{3} + 31267549962569425856578974152283531 n^{2} + 754590322553815116288148977901799172 n + 6827579385663410890567351931477081656\right) a{\left(n + 35 \right)}}{512 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(4737677638289416679609595343639 n^{4} + 723323033488004628459167356203148 n^{3} + 41401561407640373198093120710987865 n^{2} + 1052941935996603425438145037552967936 n + 10039420770762418702116379209933698484\right) a{\left(n + 37 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(6477528922882912454602034951163 n^{4} + 707716099098347993196622776029840 n^{3} + 28998741186452751910194301457314317 n^{2} + 528142681527488848118232697442650156 n + 3607321251397618916991065859636119880\right) a{\left(n + 26 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(7301676348292086109783244707997 n^{4} + 856663703648890593982996219573718 n^{3} + 37689007439533836438217876937570386 n^{2} + 736918881709161972639641978036471845 n + 5403026605242558270299664650739738612\right) a{\left(n + 28 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(8185913756073408436673452250954 n^{4} + 1153837941022339269851644855809626 n^{3} + 60977495104996851836106247879804627 n^{2} + 1431949282834147287842232675094288099 n + 12607610838737263039674935774691407598\right) a{\left(n + 34 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(9478096638003299160009932781277 n^{4} + 1149746230092757574052210483332420 n^{3} + 52297548122479333106091332009680949 n^{2} + 1057168133622924071463386008859046136 n + 8013141247760999226696287731400912514\right) a{\left(n + 29 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(11933911136070629195402527182599 n^{4} + 1541867740313376804304781300837652 n^{3} + 74693850064294164013869533490733426 n^{2} + 1607984502759780705544117957140858435 n + 12979312748452685168830805540310434580\right) a{\left(n + 31 \right)}}{768 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} + \frac{\left(20430733537045546383566237554539 n^{4} + 2799886232139713160089368306775546 n^{3} + 143864049856553224602272122787696145 n^{2} + 3284776634229296888020743180445967326 n + 28119972833901870703643271321371373804\right) a{\left(n + 33 \right)}}{1536 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(44653728311574386136752137167319 n^{4} + 5593422808232503654088888712253922 n^{3} + 262713323334424520517769797594827201 n^{2} + 5483484044883268392775177044148525798 n + 42915509512304167190972929140634305144\right) a{\left(n + 30 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)} - \frac{\left(46337294869700488078397754611923 n^{4} + 6168701371664858309379022915371518 n^{3} + 307907892955331938382140576742105933 n^{2} + 6829643069319863263530011844207377914 n + 56798785261461265829554496987613230856\right) a{\left(n + 32 \right)}}{3072 \left(n + 67\right) \left(n + 69\right) \left(n + 70\right) \left(n + 71\right)}, \quad n \geq 69\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 90 rules.
Finding the specification took 61492 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 90 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_{19}\! \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_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{23}\! \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_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= -F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{19}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{19}\! \left(x \right) F_{32}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{0}\! \left(x \right) F_{19}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right) F_{20}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{19}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{30}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{19}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{0}\! \left(x \right) F_{30}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= \frac{F_{66}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= -F_{78}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= \frac{F_{71}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{19}\! \left(x \right) F_{32}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{19}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{19}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{2}\! \left(x \right) F_{29}\! \left(x \right) F_{62}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Req Corrob" and has 187 rules.
Finding the specification took 24674 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 187 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{107}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{0}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{148}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{30}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{137}\! \left(x \right)+F_{41}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{25}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{5}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{57}\! \left(x \right)-F_{58}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{64}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= \frac{F_{73}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{73}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{56}\! \left(x \right)-F_{58}\! \left(x \right)-F_{74}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{30}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= -F_{93}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= -F_{124}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{89}\! \left(x \right) &= -F_{0}\! \left(x \right)-F_{117}\! \left(x \right)-F_{129}\! \left(x \right)-F_{132}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{28}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= -F_{17}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= -F_{109}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= \frac{F_{105}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= \frac{F_{108}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{108}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{2}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{0}\! \left(x \right) F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{120}\! \left(x \right) &= \frac{F_{121}\! \left(x \right)}{F_{0}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= -F_{0}\! \left(x \right)-F_{123}\! \left(x \right)-F_{126}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{124}\! \left(x \right) &= \frac{F_{125}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{125}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= \frac{F_{136}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{136}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{120}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{135}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{154}\! \left(x \right) &= -F_{62}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{155}\! \left(x \right) &= -F_{156}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{158}\! \left(x \right) &= \frac{F_{159}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{103}\! \left(x \right) F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{182}\! \left(x \right)\\
F_{168}\! \left(x \right) &= \frac{F_{169}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= -F_{173}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= \frac{F_{172}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{172}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{176}\! \left(x \right) &= \frac{F_{177}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{177}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{0}\! \left(x \right) F_{6}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)