Theorem Number, 1 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 1 ) B(n) t = - -------- / -1 + 3 t ----- n = 0 and in Maple notation -1/(-1+3*t) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883, 205891132094649 ----------------------------- This took, 0.011, seconds. Theorem Number, 2 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 2 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n -1 + 2 t ) B(n) t = - -------------- / 2 ----- 2 t - 5 t + 1 n = 0 and in Maple notation -(-1+2*t)/(2*t^2-5*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037, 897162524056891, 4092454235158381, 18667946127678123, 85154822168073853, 388438218585013019, 1771881448588917389, 8082530805774560907, 36868891131694969757 ----------------------------- This took, 0.011, seconds. Theorem Number, 3 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 3 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n -1 + 4 t ) B(n) t = -------- / -1 + 7 t ----- n = 0 and in Maple notation (-1+4*t)/(-1+7*t) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021, 11729463145748964147, 82106242020242749029, 574743694141699243203, 4023205858991894702421, 28162441012943262916947, 197137087090602840418629, 1379959609634219882930403, 9659717267439539180512821 ----------------------------- This took, 0.005, seconds. Theorem Number, 4 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 4 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 2 \ n 2 t + 7 t - 1 ) B(n) t = - ------------------------- / 2 ----- (t + 1) (2 t - 11 t + 1) n = 0 and in Maple notation -(2*t^2+7*t-1)/(t+1)/(2*t^2-11*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 37, 395, 4277, 46251, 500213, 5409835, 58507765, 632765739, 6843407605, 74011952171, 800444658677, 8656867341099, 93624651434741, 1012557431099947, 10950882439229941, 118434591969329451, 1280878746784164085, 13852797030687146027, 149819009843990278133, 1620303514222518767403, 17523700636759725885173, 189520099975911947202091, 2049673698461511967452661, 22167370483124807747575083, 239741727917449861288420597, 2592824266125698858677476395, 28041583471547787722875399157, 303271769654774267234274437931, 3279906299259421364131268018933 ----------------------------- This took, 0.006, seconds. Theorem Number, 5 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 5 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 2 \ n 20 t + 11 t - 1 ) B(n) t = ---------------- / 2 ----- 47 t + 14 t - 1 n = 0 and in Maple notation (20*t^2+11*t-1)/(47*t^2+14*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 69, 1107, 18741, 314403, 5282469, 88731507, 1490517141, 25037620803, 420580996869, 7064902133907, 118675936727541, 1993513514479203, 33486958228903269, 562512550385168307, 9449062742150809941, 158724968258214249603, 2666255504496087561669, 44787650571081295594707, 752341116706454253724341, 12637795210731180445092003, 212289165435439876156332069, 3566024691000523747107973107, 59901936449473006638859230741, 1006230270769646709058103966403, 16902614803900285238839839374469, 283929429980777388669488637663507, 4769434915514196847598313377889141, 80116772026295293133842353260632803, 1345798249397301355710913674409648869 ----------------------------- This took, 0.005, seconds. Theorem Number, 6 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 6 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 4 t + 88 t + 17 t - 1 ) B(n) t = - -------------------------------- / 4 3 2 ----- 4 t - 40 t - 161 t - 20 t + 1 n = 0 and in Maple notation -(4*t^3+88*t^2+17*t-1)/(4*t^4-40*t^3-161*t^2-20*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 133, 3179, 85109, 2219307, 58215317, 1525006411, 39961226101, 1047070289643, 27435930590293, 718889277456779, 18836693340854069, 493568289430075947, 12932719243854934037, 338869471651865243851, 8879224616101788878581, 232657811754582558889323, 6096214446273138377255893, 159736010124708023620466699, 4185481523915854646347591349, 109670046054998991934701643947, 2873628501997635665147304730517, 75296227771723686358192481072971, 1972948804172197365511368358465141, 51696175214587146244534510004696043, 1354568616360327657649687615830805333, 35493073304010884698238830391786483339, 930006968527597243906695648911243306549, 24368500132451252072271374037895703377707, 638515429239663191791704434697832376020757 ----------------------------- This took, 0.118, seconds. Theorem Number, 7 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 7 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 84 t + 311 t + 26 t - 1 ) B(n) t = -------------------------- / 3 2 ----- 327 t + 485 t + 29 t - 1 n = 0 and in Maple notation (84*t^3+311*t^2+26*t-1)/(327*t^3+485*t^2+29*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 261, 9267, 396309, 16072803, 661351461, 27104094867, 1112030016309, 45610618411203, 1870905530856261, 76741044139598067, 3147794134734093909, 129117222423583783203, 5296173927063613826661, 217240225442340984534867, 8910832224186253154549109, 365507492715086520702652803, 14992508471187487382276150661, 614967721768563001408332537267, 25224951489932091713515344117909, 1034685488535862023748990561260003, 42441075085175383271236192913412261, 1740862198547186990374429534226613267, 71407267328929710489173929969108752309, 2929007150387199644847336528286747060803, 120142993954684629433674553813025512171461, 4928065469040206096657480151896710739466867, 202141036008364636362264162049071751136580309, 8291488555750256285209350152189844834082327203, 340103047989190428300376269016975523901434604261 ----------------------------- This took, 0.127, seconds. Theorem Number, 8 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 8 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 4 t + 428 t + 969 t + 40 t - 1 ) B(n) t = - ------------------------------------------- / 4 3 2 ----- (t + 1) (4 t - 88 t - 1313 t - 44 t + 1) n = 0 and in Maple notation -(4*t^4+428*t^3+969*t^2+40*t-1)/(t+1)/(4*t^4-88*t^3-1313*t^2-44*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 517, 27275, 1878677, 118519851, 7683973973, 493876634155, 31830051961045, 2049658022562219, 132021241625498965, 8502934684212183851, 547653258945393861077, 35272706304598880944299, 2271815536564897071325781, 146321106461842259629238827, 9424124291372538198284912341, 606980860281183060164479614891, 39093900217050693160844280672085, 2517926216752638793065520639254827, 162172421141311245634213129716120533, 10445061488109568879466319945631575723, 672736515566836060225257039364187327573, 43329033520184218967711226313970962925611, 2790699036548630458471466949802597044011221, 179740937653265548812373690695742410216345003, 11576599355735749883642396182590947956294056277, 745615630990192859214381706164188867347020750635, 48022968757366866576428916257879650951593871234517, 3093021970593819486438920776909589845147281591475371, 199212692553680467246775012251175653142457855605667413 ----------------------------- This took, 0.116, seconds. Theorem Number, 9 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 9 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 900 t + 2819 t + 62 t - 1 ) B(n) t = ---------------------------- / 3 2 ----- 3843 t + 3653 t + 65 t - 1 n = 0 and in Maple notation (900*t^3+2819*t^2+62*t-1)/(3843*t^3+3653*t^2+65*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 1029, 80787, 9021621, 885474723, 90822302949, 9172758944307, 931406083413141, 94398513955640643, 9573580742447817669, 970699913317620118227, 98430558306938712620661, 9980744344093509575528163, 1012045611628204853726951589, 102620892480380471444257432947, 10405716631016914331840042026581, 1055134992532416485008347809239683, 106990229457513961731514562438495109, 10848762211472322934061202387162894867, 1100059735730301569471087932384318901301, 111545574632783223848680498388737920237603, 11310672358932389320477713813041748766870629, 1146897217028573971481758649385926230012254387, 116294874877351112263269879924434633023648051221, 11792249314708580187094002395478559105282972432323, 1195728909388062135031239425559656170780716632820549, 121246387061008142524912699560610429807055158889453907, 12294330479076545317047445344214926031962145650565386741, 1246639719272616513036515091981305850827022256729854630243, 126408720858262147682271038325613430980262677724568080095269 ----------------------------- This took, 0.128, seconds. Theorem Number, 10 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 10 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 4 t + 1852 t + 7945 t + 96 t - 1 ) B(n) t = - --------------------------------------------- / 4 3 2 ----- (t + 1) (4 t - 200 t - 9601 t - 100 t + 1) n = 0 and in Maple notation -(4*t^4+1852*t^3+7945*t^2+96*t-1)/(t+1)/(4*t^4-200*t^3-9601*t^2-100*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 2053, 240299, 43735349, 6681062187, 1088057350037, 172959359175691, 27743710572764341, 4435171449468008043, 709919097675516974293, 113579538904171158821579, 18174774070514805662890229, 2908096526149277167657835307, 465328371534044809464889031957, 74457106401459638377619722743691, 11913909881850461626808680785682741, 1906346720787380855109179343805507563, 305035010414351173381404364688883357013, 48808718391862705414530475574443259393099, 7809894195862974105857086191456333008767349, 1249662924243267232355768575572083435759256747, 199958847122345468509588954650878026689446178197, 31995460231498454576176979318271764704497957751051, 5119600815716756170772836215717026055556988348248501, 819188482025065051583791788374285691800587873391309163, 131078533926413992355502540702035572088890225304430265813, 20973905800745351221046575292920300267990519282550804557259, 3356039401520037612853129497838603548694742876749640306629109, 537000622174991233056792013344161411359726274149443697705086507, 85925590768339017791270337536212995058902163522913275297645108757 ----------------------------- This took, 0.129, seconds. Theorem Number, 11 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 11 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 18348 t + 86213 t - 21249 t - 153 t + 1 ) B(n) t = ------------------------------------------- / 4 3 2 ----- 107529 t + 83828 t - 24882 t - 156 t + 1 n = 0 and in Maple notation (18348*t^4+86213*t^3-21249*t^2-153*t+1)/(107529*t^4+83828*t^3-24882*t^2-156*t+1 ) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 4101, 716787, 213519189, 50799986403, 13177054542501, 3301649808344787, 838647420463231989, 211870579523572478403, 53640847906011617586501, 13569078873978509433342387, 3433617016079251515796618389, 868750687820885776080626340003, 219817129203559761272623483312101, 55618434453408471107760351188797587, 14072770737511924274122503457815598389, 3560729874921983065171825942670093923203, 900946523139154603061929759443910780103301, 227960065537494410557725527996102612495766387, 57679119517677978964299622698078931492996459989, 14594137568597271103945577419352884991623139388003, 3692650998287474053201667627586689498019378829432101, 934325249165868161050741836812027592465576087131912787, 236405681467121374078178018414249697141845758681544259189, 59816050317713009868564003698300160706246169821268271662403, 15134830331771872319307181722303580616301571821613790646978501, 3829458613236003786177814756553862890708865524474718776823188787, 968940711647408593407245161930597132118031562427087923612974259989, 245164185714407600904185592785575429983147005611548716690200067642403, 62032152468857313539165689956351070118609361140810328484091222868390501 ----------------------------- This took, 0.120, seconds. Theorem Number, 12 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 12 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = / ----- n = 0 6 5 4 3 2 8 t + 15404 t - 9582 t - 667319 t + 56447 t + 243 t - 1 - ---------------------------------------------------------------------- 6 5 4 3 2 (t + 1) (8 t - 988 t - 127318 t + 797003 t - 63659 t - 247 t + 1) and in Maple notation -(8*t^6+15404*t^5-9582*t^4-667319*t^3+56447*t^2+243*t-1)/(t+1)/(8*t^6-988*t^5-\ 127318*t^4+797003*t^3-63659*t^2-247*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 8197, 2142155, 1047947477, 388678519851, 161008621810133, 63677070942973675, 25668239857061509525, 10265399036051558656299, 4118837726072368390909525, 1650388398801475297223341931, 661668739571948876787281152277, 265212834724815816262293335440299, 106313900376753844931373966816075221, 42615575396579675931243711243354377707, 17082592526095906198196179273198418619541, 6847566537595987482079649809972730910514731, 2744858486902341164192645471370554102932972885, 1100279832825489572750346123564799443482131922027, 441048709016067021288698231175447910517785132070933, 176794956391523462492251542064993337219206231133878443, 70868497145016786588529597021212909376968182996800885973, 28407731867654827002201530755103711070773357294267999249131, 11387277474963041536390976671133008572361039978340116775512981, 4564605444064845255913709831292786988469428345987775467796007723, 1829728216950961937062951816437972416584900737780944956926854556757, 733449010126856200504021454486883674703156298603322997539043564634475, 294004019664938583094272893634465528072538414718917683962251006762125077, 117851905682497131706822758933383567825796969196130275948847089171336746411 , 47241094491855846863501403370541591990428173891511363875576678981162634\ 177493 ----------------------------- This took, 4.357, seconds. Theorem Number, 13 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 13 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 5 4 3 2 \ n 289380 t + 1565891 t - 4165856 t + 148038 t + 388 t - 1 ) B(n) t = ------------------------------------------------------------ / 5 4 3 2 ----- 2493147 t - 2327461 t - 4653650 t + 163254 t + 391 t - 1 n = 0 and in Maple notation (289380*t^5+1565891*t^4-4165856*t^3+148038*t^2+388*t-1)/(2493147*t^5-2327461*t^ 4-4653650*t^3+163254*t^2+391*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 16389, 6410067, 5164051701, 2989339844643, 1982015630077029, 1238943230838344307, 794075438414668269141, 503515382893346242274883, 320740292556839044295721669, 203912126894021324601258412947, 129746747884288549608884506323381, 82526665791586458931717457637364323, 52499920979107397664158339242362159909, 33396007648808854125676542089280613585587, 21244309399833774652133068646558828024928021, 13514048796650848715136470505328610551601554563, 8596676249488592854689267822917193897392125865349, 5468581758344603201908732982683524658764590483791827, 3478718236723772602810424098008667917415608612860972661, 2212909354051012294171433797869111585299817928695745879203, 1407693384216138121083540369076634153787220534818246296879589, 895472973314434124938482699630997254747005959032943562627265267, 569635319750237904030614671352173501845769449844348729009139666901, 362360900703098064751535262345226972956088625684683815735347125467843, 230507868028808561984115998077450396305584227923739002411626810060636229, 146632478808056618968793667721035601663393840506352266080900143852973247507 , 93277006324123094477421357444997868881992870574852455865611857337978472\ 684341, 59336103267330617445347305823872628563389034645799776952862686675\ 277742184890083, 37745348938305512708505962942160765896503880046643070641\ 796636875233055100958664869 ----------------------------- This took, 0.125, seconds. Theorem Number, 14 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 14 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = - ( / ----- n = 0 6 5 4 3 2 / 8 t + 63044 t - 2116566 t - 22761161 t + 385463 t + 621 t - 1) / ( / 6 5 4 3 2 (t + 1) (8 t - 2500 t - 831478 t + 25678925 t - 415739 t - 625 t + 1)) and in Maple notation -(8*t^6+63044*t^5-2116566*t^4-22761161*t^3+385463*t^2+621*t-1)/(t+1)/(8*t^6-\ 2500*t^5-831478*t^4+25678925*t^3-415739*t^2-625*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 32773, 19197419, 25526166389, 23093417825067, 24552667175438357, 24290782867918304971, 24796247671468698206581, 24965813676178062792344043, 25288659981903786348909940693, 25547954113686327708368259955979, 25839878893911250427748444801371189, 26121840359525377291658111082716916267, 26412772665617176235336349304356162390677, 26704331635738638010460139880791023098061131, 27000267673468135422884513316265375956477242101, 27298969537305674670167329382887311337598570615403, 27601203675571450872574654495343267841133045494152533, 27906682949265459481026335434553360799961168492739349899, 28215587917038879685750617762420009426404150796395576512949, 28527892370259342439075295313090260756955683842831575570571947, 28843662367862826251632472883522312530951764558918408666578630677, 29162923663942895640157496761696799734579981023086427452013888643531, 29485720491243647983300674135168600918663280838589669719003516741484661, 29812089505866994539539323644490035233569426933086562186796010617853352683, 301420713458709175457630759692077150028239084697838781457325777745393274\ 70293, 304757055135849042912933812802544643611266702741400713531785356108\ 19929240493579, 308130326515379586555566150691796124658154045876146892770\ 67728376105459899077097269, 311540935404585025607436749182192131645496420\ 53119682743758195459022922386989257784107, 314989295506425987654766301976\ 31157931122899214333655545420714915864895226306183651607957 ----------------------------- This took, 4.208, seconds. Theorem Number, 15 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 15 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = / ----- n = 0 5 4 3 2 16637076 t + 134208623 t - 119408756 t + 994458 t + 1000 t - 1 ------------------------------------------------------------------- 5 4 3 2 163217943 t + 31379507 t - 130790642 t + 1056990 t + 1003 t - 1 and in Maple notation (16637076*t^5+134208623*t^4-119408756*t^3+994458*t^2+1000*t-1)/(163217943*t^5+ 31379507*t^4-130790642*t^3+1056990*t^2+1003*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 65541, 57526707, 126480267669, 179092953756003, 305796712933973541, 479472944711744914707, 780715727028372364191669, 1249866274126645791838008003, 2016123639981543018567911657541, 3241172948369208763467974713198707, 5218472757764444609436917831742771669, 8396364813124258088935079513778897876003, 13513575806009305458058879558434690690557541, 21746564810480795650299839311986402134518858707, 34997517333851295190900313637309593816113655367669, 56321226495036137862248316036031945017835289927280003, 90638394268155328363254754993440623428856311806208913541, 145864619485588244544019352123124157700524068214878849574707, 234740902084299059411038797810254011395117214677861978962139669, 377769688208986558547510000919771272763230010446650320157033740003, 607946909520853126875362736087098101024728776862880457427107091365541, 978372211655828246298777400622103094630092237156940902130720642375426707, 157449978199775238494422027228259851463768072131227587790706844323894244\ 7669, 2533851051284193965177543432951745435376305942722004233034682819661\ 665099176003, 40777403380675195562595971869148831188331064530432151817431\ 73514632292687722553541, 656232974452776450194231041199429473684673228697\ 2810041363631644116015474427728094707, 1056079305468033120983073037028725\ 1566451896809961509357907274672369735619389640133251669, 1699554184177104\ 248360257194945417104677593163940852081431441914677651688663409012996870\ 8003, 2735101816891187562158855402786045017959638449912852201562122649137\ 8566204498064865777232717541 ----------------------------- This took, 0.145, seconds. Theorem Number, 16 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 16 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = - ( / ----- n = 0 6 5 4 3 2 8 t + 255692 t - 44606766 t - 585266591 t + 2559407 t + 1611 t - 1) / / ((t + 1) / 6 5 4 3 2 (8 t - 6460 t - 5368054 t + 635116787 t - 2684027 t - 1615 t + 1)) and in Maple notation -(8*t^6+255692*t^5-44606766*t^4-585266591*t^3+2559407*t^2+1611*t-1)/(t+1)/(8*t^ 6-6460*t^5-5368054*t^4+635116787*t^3-2684027*t^2-1615*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 131077, 172449035, 627876198677, 1393629282910251, 3826423387861508693, 9521438606497700756395, 24762233053988928979461205, 63116586434155371090755063979, 162348584233439466109903919059285, 415872726889538145911439369671402411, 1067296166763527092065861378498019144277, 2736786964479305961330725102224995434301099, 7020435797617251208342094787627771278299999061, 18005758439278395169086799754833026762859700060587, 46184165497728409893033251294372665092731648466963541, 118456587149437110835439731402752129198264089551533388971, 303831213653355940658624779530941202998974893237245502854485, 779295846141847525953614135378002287056781761498055838800371627, 1998820453297334406947202043813479317426663560807156741139084789333, 5126778455783584523413513780519981447151436133821566869097561476219563, 13149693027951844040596026810938823567737794246180346456001714449251180373, 337276857976997353728034560783305857959940307839190559502772808055789041\ 16651, 865082513613026488152776756514700529817289454614714079670422804882\ 28027845369941, 221885282011933838149132511130688622982837821943466989043\ 691394476948494459526787243, 56911426397731100172368912904219207864793466\ 3254932262057309548069257354666931014814037, 1459722962544546701200712724\ 168436397071401233412814182817575513576086661790936233717119915, 37440480\ 320968473573119089236946692137991003986377522631916845339664934588417670\ 25272751626837, 960312058414555541485133752178754291332004762003098449507\ 4872088266680839135023455167072276283051, 2463107424799599765034063849150\ 0785276550629788704516197612449018733055787428741809762357305903289173 ----------------------------- This took, 4.195, seconds. Theorem Number, 17 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 17 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 6 5 4 ) B(n) t = (1347896340 t + 15329386299 t - 15022392733 t / ----- n = 0 3 2 / 6 - 2764163954 t + 6569850 t + 2599 t - 1) / (17963099343 t / 5 4 3 2 + 1846942698 t - 19072608751 t - 2949663284 t + 6824193 t + 2602 t - 1 ) and in Maple notation (1347896340*t^6+15329386299*t^5-15022392733*t^4-2764163954*t^3+6569850*t^2+2599 *t-1)/(17963099343*t^6+1846942698*t^5-19072608751*t^4-2949663284*t^3+6824193*t^ 2+2602*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 262149, 517084947, 3121511196981, 10877537629760163, 48074921446464884709, 190104093110960686755507, 790578784916366700719493141, 3212380731914884719860977907523, 13192016916808564419370482498364869, 53911967214624328036406025804647594067, 220813288644984097108890484642961802259701, 903488571377809861858276847527924083039667683, 3698476010911945611088164553854681049702518776229, 15136661897198851554222532202411523671071446803338227, 61955509908685831837453327923995490825612092614644834261, 253577248573254935701960437700959007997493734005230873114243, 1037885764064131628571900289357221888169865424323902725853504389, 4248002261165529300081289518390364064045163516036726230360176861587, 17386885524527060567899370244617096437326863712333891473038259958794421, 71163613633253118026185643660403010400464130203321146559615034147161024803, 291269214142243900051709703760299417662777022560487419678679411784782722\ 222949, 11921502527904305643174979951519655210740542551858378460242823439\ 39853933869629747, 487941198633438626405255757954416415681864838126955974\ 3957775537144366210673932453781, 1997119004819660357850107579231130481563\ 0622310832161963886381624360759569023969114176963, 8174108866498441799279\ 0331891393132093400367349808459696020471563997119129182624303017949509, 3\ 345622087751543211170393170393990356159720357217409637006587718601736650\ 39779071263407691076307, 136934648538128084856446504102372755225713649278\ 2583108527423264542613080222374221473956775917517941, 5604667056460074279\ 329395072474900643077694508242117864016343382956232941278849005431408227\ 216168244323, 22939623538754942003604938271875377414822767657488673874503\ 504620462532625952640194216052733279261603941669 ----------------------------- This took, 4.315, seconds. Theorem Number, 18 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 18 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 8 7 6 5 ) B(n) t = - (16 t + 2063568 t - 1371988544 t - 45547388948 t / ----- n = 0 4 3 2 / - 275807280985 t - 12680716224 t + 16841706 t + 4196 t - 1) / ( / 8 7 6 5 4 (t + 1) (16 t - 33600 t - 69396808 t + 26801802800 t + 295272544809 t 3 2 + 13400901400 t - 17349202 t - 4200 t + 1)) and in Maple notation -(16*t^8+2063568*t^7-1371988544*t^6-45547388948*t^5-275807280985*t^4-\ 12680716224*t^3+16841706*t^2+4196*t-1)/(t+1)/(16*t^8-33600*t^7-69396808*t^6+ 26801802800*t^5+295272544809*t^4+13400901400*t^3-17349202*t^2-4200*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 524293, 1550730539, 15536767906229, 85131233595355947, 606165709691933872277, 3814190351870299689338251, 25390667930742128137814624821, 164665659950123502950378459489643, 1080811023436294564067945551474209493, 7054840016012511872727221659939850203979, 46167371294356578327249080985964886645080949, 301766340065707522801249794074939817964757782827, 1973525324988724802101310599853271140936339942322197, 12903444035408013504596002298182837890047907662143420171, 84375981348707692466092539766572259832831688735779193108661, 551707455887447038684016454680092146310858815533178723004740843, 3607526741922485696494085583422701769285825679390268867214023642453, 23588772120938425983564811881827269742898623123217851294319784956689099, 154242261599394919396580770167961826914114827292799848318681624703075868149 , 10085568546833038628516887950255936453528424305601560935583987966823673\ 59973547, 659474291536655448278767455135330158586586635780885075137492779\ 0932016710996023957, 4312162629205706482986085849189211762816629097703068\ 3870215061257997846784373750390411, 2819632423001769295939748336388695342\ 50991414271011215719185364509161904347459991935693621, 184369811991994043\ 430474812170165468943015343524431129554748431488671305681400140460211965\ 8603, 1205555341520284704824527717608016786447492218553742465988905230768\ 8047625556360757467399541282773, 7882872103579176027339171109170316740892\ 5486800772948509155684360577776330490057555686346732253684299, 5154443858\ 962235233518261104000908430985529235390745623778911934980387618811868782\ 45523888553203467303029, 337038214178332410856160237131780782657486055652\ 2337962443719455413232110621196283909053365607593483955134507, 2203821812\ 473034130827368401824570449782182747558824680647118869835810559323674082\ 8724756543063720569092352558357 ----------------------------- This took, 195.123, seconds. Theorem Number, 19 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 19 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 7 6 5 ) B(n) t = (75808868436 t + 1490826289911 t + 1820299893334 t / ----- n = 0 4 3 2 / - 3287973427007 t - 57102085832 t + 43115177 t + 6782 t - 1) / ( / 7 6 5 4 1428411421143 t + 3593918815161 t - 1326810706493 t - 3552060363299 t 3 2 - 59697995099 t + 44143403 t + 6785 t - 1) and in Maple notation (75808868436*t^7+1490826289911*t^6+1820299893334*t^5-3287973427007*t^4-\ 57102085832*t^3+43115177*t^2+6782*t-1)/(1428411421143*t^7+3593918815161*t^6-\ 1326810706493*t^5-3552060363299*t^4-59697995099*t^3+44143403*t^2+6785*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 1048581, 4651143027, 77402758177749, 667883008596933603, 7667018822366372188581, 76866039504711793631436627, 819838157713097537719834175349, 8495652343427962472859799989720003, 89217464291693987735006352934376301381, 931151763059358275450579930211845331039027, 9746141564679871432180554269387383180912681749, 101875496143768695038682103266710270164105340983203, 1065548285644841430450830298183812059788899409350730981, 11141743463223225292193465281586334091759076745438687550227, 116517273799361560380641260335126794001756384427119176452064949, 1218431197374391039808201556940759995366962719446441113123388643203, 12741599874344895234149556826555489529550123338865271370978850672117381, 133242029321928783729595636182562722696164076796779888296824527187426666227 , 13933529562488605252277618509646793788939022643736596257546116644037333\ 10308949, 145706803424742542897483244062551611244659183079325580087841213\ 40271058551086556003, 152369864631093850702396372160800440275681499894426\ 733812598428588897704756415513196581, 15933752672782771959362121467017774\ 31717205022062499194329107986856740420015900991278819027, 166623856737256\ 751762676931185256429073664429889527117265904428190385852132530836382893\ 85653749, 174243360133596531598961124078788531658307801587606095809169446\ 649492294782580609237887414467873603, 18221130490620608163140577770706733\ 27864728491432167754657754807237385066676082747290415751254451120581, 190\ 543609188054849772886208544223908255425556574462320667038197809647232621\ 00265629045372595392257767336627, 199256942297842420043003828423424935832\ 696442056811342433668863160903649741760474066513820876202287507903955349, 208368724552920715879104855758319684208666935190090554058637403132162422\ 6022893996965380362647690122625132268724003, 2178971783390455411619022822\ 368803639763154117576432018613933260574649847198491720202373996292681615\ 8384838665512897381 ----------------------------- This took, 6.033, seconds. Theorem Number, 20 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 20 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 8 7 6 5 ) B(n) t = - (16 t + 8300880 t - 18550240640 t - 2366125043732 t / ----- n = 0 4 3 2 / - 34979269477849 t - 252584574960 t + 110369322 t + 10964 t - 1) / ( / 8 7 6 5 (t + 1) (16 t - 87744 t - 449690440 t + 524164974416 t 4 3 2 + 36830725406121 t + 262082487208 t - 112422610 t - 10968 t + 1)) and in Maple notation -(16*t^8+8300880*t^7-18550240640*t^6-2366125043732*t^5-34979269477849*t^4-\ 252584574960*t^3+110369322*t^2+10964*t-1)/(t+1)/(16*t^8-87744*t^7-449690440*t^6 +524164974416*t^5+36830725406121*t^4+262082487208*t^3-112422610*t^2-10968*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 2097157, 13951331915, 385895673640277, 5251177395930795051, 97244672945975981949653, 1555280304570457849232350315, 26600359864111765544436395064085, 440921887482572404558750489400201259, 7415319819645138407280213730035651548245, 123872046552171296959780858440863254325261291, 2075740598392489501192571510842928353147672029077, 34733076713923651914033301677879129580055621533878699, 581574755257430159174128812571706074183654970356467673301, 9734917493128780395787962624185419456933025794620427344431467, 162975344730293489742103577387845940604754060497333044960004536341, 2728238609859811663774647889881036583697745109950507430066070848717611, 45672663465488015725838915395390088957668662306900424790815641431852524885, 764581950034306013076629831889904793742943694899949893494726676911447599\ 867627, 12799548794576688344133522292038771504188490256227326133013321150\ 433806407660355733, 21427126122970625042857200914134871799826427296315752\ 7971022819467320252891677243303083, 3587020177580038990964615484273572422\ 304334227679004266222542329428437125752003021252450773, 60048674028118237\ 815168587295644339552458489638286861296412826296044050707406393603672790\ 232171, 10052478650785934260993108888609243427459650539824890337513719889\ 16040968717758398625520654312014101, 168284003029629705697509455866708170\ 75367781044895921810977126392412704210789507043060349322338331307563, 281\ 716664701629379197626885987865614052870480923362493038474189404354532950\ 686908802613749425382611182700117, 47160915637867076084492479673757496489\ 445056414906421916486622363517082013660494305073876702679389223841481589\ 55, 789499619511730499340121653406552871480793691085266372265642887739694\ 42491033026993030812619712997544841907544635797, 132166569838990555032362\ 711070880838077653819854553121539123347789072835199913644962353780579063\ 3722912931652480017637291, 2212540931701335286384281021235094060047358379\ 357006302918506456128970823254303222540231505036206250415371929719532826\ 9642453 ----------------------------- This took, 209.763, seconds. Theorem Number, 21 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 21 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 7 6 5 ) B(n) t = (24228053037540 t + 491312623390091 t - 178299513045558 t / ----- n = 0 4 3 2 / - 337018569789879 t - 1102832042704 t + 282555981 t + 17730 t - 1) / / 7 6 5 (524736161161947 t + 352323870537425 t - 524479896911361 t 4 3 2 - 353204113242387 t - 1136220014047 t + 286697091 t + 17733 t - 1) and in Maple notation (24228053037540*t^7+491312623390091*t^6-178299513045558*t^5-337018569789879*t^4 -1102832042704*t^3+282555981*t^2+17730*t-1)/(524736161161947*t^7+ 352323870537425*t^6-524479896911361*t^5-353204113242387*t^4-1136220014047*t^3+ 286697091*t^2+17733*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 4194309, 41849801427, 1925024514265461, 41367612415896917283, 1236438768446007597445989, 31583707719635071645363121907, 866874629420667676922943674013141, 23007766234396116866340390788203882563, 620204375367704352703224138161888782723269, 16598227390636830743768378285051654279178637587, 445699071838159194602269314506045560910054956653621, 11949429633148177164506310001858135991995402803748518243, 320601554079835985439982697955102362741592926605940553408549, 8598818980302297395075637497227192994033481456055495018947522867, 230663777346179920742669196474599710397794818227438277580446699185301, 6187122427636298847060273947119570237258267474431357183910746732990318723, 165963481696217801471259713969801918624628693275889630440055331779258628\ 004229, 44517383158690753194201686646773969105822301088674261322005342009\ 62160824692920147, 119412514733474850434973987504185019533580811619676903\ 170414386790499958296374042062581, 32030860812513871459687073030356854260\ 17544948441695431931800545799305016258754234396202403, 859187696807992074\ 262073841703769662435795848028316439878097168569432270996492079015056759\ 24709, 230466165597254481753907240889480853593522595425337029257451795833\ 8245759978204466435133907166931827, 6181963663674680998960192338678597466\ 6327736967183626557321791257848963459820137935846790189962658347861, 1658\ 233377485521618302150080437069696212021158445369756937998139961684799545\ 505985423044884071240597131583683, 44480011231492214783280125931934109275\ 228512378541663984364217824598405553521035190650344072640393326643129848\ 389, 11931199552464100985938777256043831691582568798988597996168336050405\ 55647178262754412281293686225945659751898406012307, 320039318839832486569\ 130327549914127529349261570944157693037485804183097239922098126262085555\ 73930704191528972807533103541, 858464938223625789856728005697964440780279\ 506776519173048471207154284131124721146251459960456519679663248543100390\ 456872724963, 23027234756297551163164058163224639211210439047806031332190\ 864929806448864953066408553126798983513838510942191531499055977844789669 ----------------------------- This took, 6.760, seconds. Theorem Number, 22 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 22 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 8 7 6 ) B(n) t = - (16 t + 33325008 t - 233016526784 t / ----- n = 0 5 4 3 - 123377166461588 t - 3036610091030425 t - 4764905230944 t 2 / 8 7 + 723669546 t + 28676 t - 1) / ((t + 1) (16 t - 229440 t / 6 5 4 - 2927773768 t + 9765738214640 t + 3150339474879849 t 3 2 + 4882869107320 t - 731943442 t - 28680 t + 1)) and in Maple notation -(16*t^8+33325008*t^7-233016526784*t^6-123377166461588*t^5-3036610091030425*t^4 -4764905230944*t^3+723669546*t^2+28676*t-1)/(t+1)/(16*t^8-229440*t^7-2927773768 *t^6+9765738214640*t^5+3150339474879849*t^4+4882869107320*t^3-731943442*t^2-\ 28680*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 8388613, 125541015659, 9607362981882869, 326455224911603542827, 15755354857110628664043797, 643503345476299490880193939531, 28363400023583844056090146649571061, 1206510585920530029215750781185408196843, 52171350918667177363221527870686644411700693, 2238849831516433321802303512791335360791095389579, 96416103720424125506668566868292412774327562569123509, 4145378510612435943265034610870260820805545911523840978987, 178364222349441371064592908426676427550001254010390157642458517, 7671828160615812463083708059541737935442864509433923070590957764811, 330035470335518705915894518056617331680851537471127778184075831533474421, 141967730978732249048340542901746421370236350233975110707283720509817115\ 31883, 610708309973277604142278022303867706373891157553186079799819247820\ 690260249746773, 26270660429322061092168109834673744522717207483151159089\ 553382089711226653961686002699, 11300857749443340381350786499070931270411\ 16239014247488072951555074301247637040325198992949, 486127642453733811831\ 974546929024939041979183081741354002423682485660212155129329502329192055\ 47, 209117281555088318218903862809658041798094663176936745041219301713448\ 5107537091919600759418218614037, 8995580791324930936654359995367532461248\ 6912948956110610388377557847574077152642277065472439776831311691, 3869622\ 880544611108723315729138620050838855700050924183769694756349293525233937\ 738863106816585293057511035381, 16645927802559184228430327112244541949036\ 860563411412839919320260956470333995894464818652430070472986505867969892\ 3, 7160566843973769368100281625576699048512678255658470940375319891328211\ 228477246180095634563190543268978688221731927253, 30802558062257878583547\ 005124123244858130144738595006498454618135866383935434329435833212743973\ 7599133193276684687102241419, 1325031410181488113205296862827747870987405\ 335250187915475423664561789235613034015022899533419825528611459583102975\ 5446028158389, 5699877982393364757622808214323355724826996850478491663469\ 098494415899426860225726553454536936466387081074930565378288614543920817\ 07, 245191237548647125981445811229333345507058882555082832266412215836062\ 99200238918466344794150209388441629549942594200064915790415288439957 ----------------------------- This took, 218.557, seconds. Theorem Number, 23 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 23 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 8 7 ) B(n) t = (7586660581874892 t + 214666561582310301 t / ----- n = 0 6 5 4 - 1761279457237853 t - 246398742959564703 t + 25926856168486983 t 3 2 / + 20423908865911 t - 1854356343 t - 46389 t + 1) / ( / 8 7 6 221507240638423761 t + 229634216915976744 t - 248226783642364532 t 5 4 3 - 229614570190537800 t + 26760022845792246 t + 20831244415960 t 2 - 1870994388 t - 46392 t + 1) and in Maple notation (7586660581874892*t^8+214666561582310301*t^7-1761279457237853*t^6-\ 246398742959564703*t^5+25926856168486983*t^4+20423908865911*t^3-1854356343*t^2-\ 46389*t+1)/(221507240638423761*t^8+229634216915976744*t^7-248226783642364532*t^ 6-229614570190537800*t^5+26760022845792246*t^4+20831244415960*t^3-1870994388*t^ 2-46392*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 16777221, 376606269747, 47965943992785429, 2580276835382669551203, 201154079837074246060131621, 13150359107245781713777590662547, 931395753504504383046272110320223029, 63554232830778674988349504605964386014403, 4411723557287859352651451154610489851215550021, 303824301918898859556329589929408252246295328975347, 21000492121869488343439432283147456776721563393163116629, 1449106203771336199728909579089130533788013421548147533773603, 100071656387861553384031431476698302031318479971470095731670680421, 6908198141523049339330465319734767210680198538130944003654851719704147, 476970046280273465863995628513241529919952366755992719314385655367449338229 , 32929403914666081305584465974308395969766536384872980203923129812932677\ 347772803, 22734852352136890098293535464188627989395283181288272835390666\ 04611089258706788850821, 156961536781582455723699672979371419222394361173\ 222509104229705718181144374364744100704947, 10836715126151134720243145239\ 701655992863653069003786851993553502085821790929124028057437959829, 74817\ 043389171929429279459181458437404290337111837622787788919228319048635635\ 0565288498726388396003, 5165402086224171889609435734681550352486998115351\ 1204749353444723866325175406985675854384586231694029221, 3566213985184387\ 692388339093228347314924526970915283051787572568214132084074771573115751\ 171041205364499113747, 24621290647524103324769450207663718695992315576401\ 7088858918822104060355450675269867300033148106925614253759893429, 1699863\ 940664119117918067937831893208137886617193216469942302247892988271423973\ 9094464255106926687782331646144622747203, 1173593079471424157575358682467\ 375077863280353195553825035462130486712558913262314492681706699829459266\ 944309386089446183621, 81025347736134975967942451957272772807036319834084\ 707585263061608418044676998316576476091878379630799390268367421833588692\ 466547, 55940233413082412843894020938679814736628533525547846443858828600\ 93230694768628240394737282240157111651479400102561950180188960931029, 386\ 213670022767380537477168533831517675014175453479969982849572349947988701\ 543074222274505297442525702826773708848771286989812108865930403, 26664350\ 560623451406872835969668317961025468208542011745860608038809460242629442\ 115686496698897711272105935914893645213046225155618451498242021 ----------------------------- This took, 210.913, seconds. Theorem Number, 24 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 24 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 10 9 8 ) B(n) t = - (32 t + 267234672 t - 5592343256544 t / ----- n = 0 7 6 5 - 11490059901791704 t - 168176089283202742 t + 7969097795834637895 t 4 3 2 - 213225285448324885 t - 86950650711878 t + 4755292108 t + 75047 t - 1) / 10 9 8 7 / ((t + 1) (32 t - 1200816 t - 38308370752 t + 353433121457848 t / 6 5 4 + 437590012869780314 t - 8232940590838251851 t + 218795006434890157 t 3 2 + 88358280364462 t - 4788546344 t - 75051 t + 1)) and in Maple notation -(32*t^10+267234672*t^9-5592343256544*t^8-11490059901791704*t^7-\ 168176089283202742*t^6+7969097795834637895*t^5-213225285448324885*t^4-\ 86950650711878*t^3+4755292108*t^2+75047*t-1)/(t+1)/(32*t^10-1200816*t^9-\ 38308370752*t^8+353433121457848*t^7+437590012869780314*t^6-8232940590838251851* t^5+218795006434890157*t^4+88358280364462*t^3-4788546344*t^2-75051*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 33554437, 1129785254795, 239546738545360277, 20422833293449519198251, 2572667327642686810236253013, 269464086303180854559288857563435, 30685957108511093930339586180065566165, 3361570129468633340305696764397611286964139, 374858225600483937264906434047990282753921816405, 41460224425774527292969816644960372110791886971414571, 4602923017429608958769878620850453650022790098419104958677, 510131111159474895937360681873588614440450600753903323120125099, 56581816644380962051782256074094121282592330495386632370944887993941, 6273533821924716937937388643429758699414435857478074024759978994966914347, 695698575124320053037079170384396767152182589594236718822361104285754356\ 649941, 77142932945823328251200726024701225612523213394571921549626894627\ 019024295343540651, 85543448495813121956299841167179665972553857886987262\ 21739938796076785218125633176352085, 948571763040669317146160157566156565\ 110589474768911546113341924572248254041799814876713194027, 10518575552076\ 456663406164705809049774365120143800327055303306004473102745563484720226\ 9765148690133, 1166385602409734095265288448793460043945445910062032017071\ 4866591567077930733604408828256111475777593003, 1293385740896274147222394\ 032939951305162874122372263953772653111821108624379421007038694189151015\ 257429337173, 14342130354686924154521046292726501123351404122337583398915\ 1408223256407918119633821974986857195035356121283579691, 1590374524421395\ 990312488431129042722923074289338118103293170082780181441796708987309837\ 9215058383803319468458222677461, 1763539075694879557303678434914149254192\ 407004915343914588766199438422178304077673950351775888288223578271210549\ 729823392683, 19555584225231915257441015888937435525896785966189117820759\ 3106680582400847519875213415083746517777936763504233291547614562566997, 2\ 168485331249584749447791829933721294246925930365929069952286386489231048\ 4726969348917668822246827486707915403685866464436037232663595, 2404596380\ 072836054971370889432177564752632430765711101503236545514089713054203975\ 520043726995346118099709720970281544319084014368947487957, 26664158742504\ 794538200385844698487558933535415810704860270112477395078177093708287166\ 6104911203733877286936025231688318345039585662950465321131, 2956743051112\ 445025804473531870652073337759499579842231423909902510991329269991016720\ 5887302619878141808822220303806066792820923277105753677804917333 ----------------------------- This took, 208.170, seconds. Theorem Number, 25 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 25 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 9 8 ) B(n) t = (1514546390971357380 t + 54170017565147893899 t / ----- n = 0 7 6 - 199286356058564027268 t - 44752551120091153312 t 5 4 3 + 190056893562569444100 t - 1700158252059894306 t - 368201661526540 t 2 / 9 + 12204424632 t + 121416 t - 1) / (61308282084557768883 t / 8 7 - 142442509995905277897 t - 254968183731662520852 t 6 5 + 144181998196664158508 t + 193662298543101511578 t 4 3 2 - 1737464314515507006 t - 372997478143876 t + 12271169244 t + 121419 t - 1) and in Maple notation (1514546390971357380*t^9+54170017565147893899*t^8-199286356058564027268*t^7-\ 44752551120091153312*t^6+190056893562569444100*t^5-1700158252059894306*t^4-\ 368201661526540*t^3+12204424632*t^2+121416*t-1)/(61308282084557768883*t^9-\ 142442509995905277897*t^8-254968183731662520852*t^7+144181998196664158508*t^6+ 193662298543101511578*t^5-1737464314515507006*t^4-372997478143876*t^3+ 12271169244*t^2+121419*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 67108869, 3389288655507, 1196603273645385141, 161847861705193875972003, 32954332192850591013398320869, 5535138788246895757542569614707507, 1014011960833117017593747204348833077141, 178470087183489813891051852650445241285704003, 31990953461170858532389303238047150609396887292869, 5686511594838048926189148958556953900978419897618519507, 1014687325445049294309637401320976216422261741600283266529141, 180740032560999023035344077974567908359424131821430075197973196003, 32220070852538740569980464381083946296393770452883989624276597318024869, 574167058256813535645172210065923663030114754957088803806610144915928409\ 1507, 1023348384854286676450902535979001934179596533197328729835604587481\ 412725969741141, 18237911097971887014902080267324729933325726205281200635\ 9515265672038596967055362448003, 3250439563389568515641367893527631721591\ 4079145357940185509986242214968166981919333794516869, 5792980076157235630\ 630184285894431013105264464450695708477466370741843117458800656261236275\ 423507, 10324409200526133850612965093205842697222726953364459738701581498\ 59208633435546415113350555212046713141, 184003846401267741624737075232686\ 494127307725915121711190740195535658541238701304891657725210156699917460\ 003, 32793613423874450655164298463919271071831365107039410310305463001125\ 231837354699287815534535277324710533420768869, 58445534282834250453275947\ 117408505659630393549115695348002650863257789412023368835297877772042666\ 92799656320976515507, 104163013511005939066866772935986041765877939721589\ 470422720098540900299573492849169727167215084690028536633421815340144514\ 1, 1856417583715254244803284767281563539747085443415996387480010577979824\ 91773462410494620401998259513948337491788314925598902232003, 330855105493\ 602026926560627535356427738032477448989426830250282403486190662372990426\ 92231491074024935245400042639528982903975780780869, 589657727546595304542\ 955478731065371307551345759606978181553466077899220774121283058902184561\ 6458169146496571560904097852073383126731367507, 1050901844660074638853577\ 819172836412326069259798579057880428379520269906790822272998277294358057\ 605991278771485931731348597535915783045457937141, 18729419258120919272161\ 983278793110485762393027201468881059576354003007026054010481494503973204\ 9777143984485969280325042479284983845750354889340764003, 3338001056829689\ 973777186845695478303801653780855939174717399112760055672486103367662542\ 7633128376154589520689840852846579508331782016339327021820618552869 ----------------------------- This took, 199.348, seconds. ----------------------------------------- This concludes this article that took, 1272.789, seconds to produce.