Rational generating functions for the Certain Stanely-Stern Sums By Shalosh B. Ekhad 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, 0.032, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) / ----- k = 0 Then infinity ----- \ n 2 t ) B(n) t = -------------- / 2 ----- 2 t - 5 t + 1 n = 0 and in Maple notation 2*t/(2*t^2-5*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 2, 10, 46, 210, 958, 4370, 19934, 90930, 414782, 1892050, 8630686, 39369330, 179585278, 819187730, 3736768094, 17045465010, 77753788862, 354678014290, 1617882493726, 7380056440050, 33664517212798, 153562473183890, 700483331493854, 3195291711101490, 14575491892519742, 66486876040395730, 303283396416939166, 1383443230003904370, 6310649357185643518, 28786360325920408850 ----------------------------- This took, 0.016, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) a(n, k + 2) / ----- k = 0 Then infinity ----- \ n t (4 t + 1) ) B(n) t = ----------------- / (7 t - 1) (t - 1) ----- n = 0 and in Maple notation t*(4*t+1)/(7*t-1)/(t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 1, 12, 89, 628, 4401, 30812, 215689, 1509828, 10568801, 73981612, 517871289, 3625099028, 25375693201, 177629852412, 1243408966889, 8703862768228, 60927039377601, 426489275643212, 2985424929502489, 20897974506517428, 146285821545622001, 1024000750819354012, 7168005255735478089, 50176036790148346628, 351232257531038426401, 2458625802717268984812, 17210380619020882893689, 120472664333146180255828, 843308650332023261790801, 5903160552324162832535612 ----------------------------- This took, 0.009, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) a(n, k + 2) a(n, k + 3) / ----- k = 0 Then infinity ----- 2 2 \ n 4 t (4 t + t + 3) ) B(n) t = ---------------------------------- / 2 2 ----- (t + 1) (2 t - 11 t + 1) (t - 1) n = 0 and in Maple notation 4*t^2*(4*t^2+t+3)/(t+1)/(2*t^2-11*t+1)/(t-1)^2 For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 0, 12, 148, 1648, 17880, 193460, 2092380, 22629368, 244738400, 2646863804, 28626025188, 309592549632, 3348265995752, 36211740854212, 391632617405036, 4235535309747208, 45807623172409456, 495412784277009868, 5357925380702289908, 57946353619171169552, 626694039049478285560, 6777741722305918802388, 73301770867266150255484, 792763996095315815205912, 8573800415313941666754432, 92726276576262726703887324, 1002841441508262110409252100, 10845803303438357761093998880, 117298153454805411151215483912, 1268588081395982807141182325732 ----------------------------- This took, 0.245, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) a(n, k + 2) a(n, k + 3) a(n, k + 4) / ----- k = 0 Then infinity ----- 2 4 3 2 \ n 4 t (4 t - 55 t - 69 t - 21 t - 3) ) B(n) t = - -------------------------------------- / 2 3 ----- (47 t + 14 t - 1) (t - 1) n = 0 and in Maple notation -4*t^2*(4*t^4-55*t^3-69*t^2-21*t-3)/(47*t^2+14*t-1)/(t-1)^3 For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 0, 12, 288, 5196, 87952, 1478860, 24843328, 417321356, 7010147120, 117756179084, 1978063442016, 33227428630476, 558152982632400, 9375830902523212, 157494822819091840, 2645591571885928204, 44440538678900370800, 746510345383243884812, 12539850153273731892128, 210643888378844709162828, 3538387394507691327306768, 59437686276913379913055692, 998431815418648811166317248, 16771616670876012212242190476, 281728928716940665096207721392, 4732470985568341885322291209100, 79495853447652997654033840003808, 1335368084588854035766621447066316, 22431458296283647390472290739307600, 376802716123647203147643278362638924 ----------------------------- This took, 9.606, 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 ----- \ B(n) = ) / ----- k = 0 a(n, k) a(n, k + 1) a(n, k + 2) a(n, k + 3) a(n, k + 4) a(n, k + 5) Then infinity ----- 2 5 4 3 2 \ n 4 t (176 t + 56 t + 1407 t + 444 t + 75 t + 2) ) B(n) t = --------------------------------------------------- / 4 3 2 4 ----- (4 t - 40 t - 161 t - 20 t + 1) (t - 1) n = 0 and in Maple notation 4*t^2*(176*t^5+56*t^4+1407*t^3+444*t^2+75*t+2)/(4*t^4-40*t^3-161*t^2-20*t+1)/(t -1)^4 For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 0, 8, 492, 14184, 379104, 9932128, 260348044, 6821338160, 178738901104, 4683388182680, 122716539639884, 3215478561995608, 84253554695130336, 2207653070418744624, 57846011990614703852, 1515709664423272441568, 39715370327552389463856, 1040641672390537570515240, 27267405073075654558854764, 714473962690834042526049480, 18720998275996172527767911776, 490536807149381195434134377088, 12853286754310348744736139603148, 336788143072446430924559052585744, 8824688617185766029520953846812016, 231228832701323012394280793466922808, 6058771633969249194480492856240373772, 158754915136413129740546204790666964920, 4159774390350895892553936593591354750304, 108996455094008396415036313434200252284240 ----------------------------- This took, 11.180, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) a(n, k + 2) a(n, k + 3) a(n, k + 4) / ----- k = 0 a(n, k + 5) a(n, k + 6) Then infinity ----- \ n 2 8 7 6 5 4 ) B(n) t = 4 t (12 t - 540 t + 7004 t + 4233 t + 11383 t / ----- n = 0 3 2 / 3 2 + 21540 t + 2256 t + 191 t + 1) / ((327 t + 485 t + 29 t - 1) / 5 (t - 1) ) and in Maple notation 4*t^2*(12*t^8-540*t^7+7004*t^6+4233*t^5+11383*t^4+21540*t^3+2256*t^2+191*t+1)/( 327*t^3+485*t^2+29*t-1)/(t-1)^5 For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 0, 4, 900, 40944, 1768064, 72064708, 2962683076, 121451889552, 4982580563888, 204367817345972, 8382933073725604, 343852754442261648, 14104280695926471936, 578533945201578285956, 23730500399072878438948, 973385574783447106452976, 39926654962351228508253552, 1637724871308654304986741044, 67176746007645507875406234692, 2755478212979105918028159118000, 113025126023020072900120070753280, 4636102383906948565422398018211780, 190165196630110411389002986050044420, 7800259574677599548349288898850801104, 319953653510791508606688950454625649584, 13123965864829635635067588655567211219508, 538323216913712890143514429451869228043876, 22081121579638075920484086238852970531093072, 905730822850454244266310100701263202455371648, 37151569520718424020234703964611084820295586692 ----------------------------- This took, 843.069, 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 ----- \ B(n) = ) a(n, k) a(n, k + 1) a(n, k + 2) a(n, k + 3) a(n, k + 4) / ----- k = 0 a(n, k + 5) a(n, k + 6) a(n, k + 7) Then infinity ----- \ n 3 ) B(n) t = - 24 t / ----- n = 0 7 6 5 4 3 2 (12 t - 1806 t - 9264 t - 53812 t + 68349 t + 34729 t + 1335 t + 57) / 4 3 2 5 / ((t + 1) (4 t - 88 t - 1313 t - 44 t + 1) (t - 1) ) / and in Maple notation -24*t^3*(12*t^7-1806*t^6-9264*t^5-53812*t^4+68349*t^3+34729*t^2+1335*t+57)/(t+1 )/(4*t^4-88*t^3-1313*t^2-44*t+1)/(t-1)^5 For the sake of the OEIS, here are the first 30 terms, starting at n=0 0, 0, 0, 1368, 97704, 7071864, 444927384, 28886025168, 1855830938832, 119623102988352, 7702662857640432, 496145497719577704, 31954517641757916696, 2058115170582614805480, 132556979162411651350632, 8537622315091570486069344, 549883681821216303706552320, 35416436881554359307495485808, 2281071277555460224854391031808, 146917273451432130332781181061304, 9462521066204182508630034873867144, 609453899561391771789013184945184216, 39253181336405945398176201257406042552, 2528185063110701156210965279107969645168, 162833163964648994840275223171866341992880, 10487618044450911868705059580688802101022240, 675477461514252648526268961814724279384845968, 43505570004679800487506896925070273267387964040, 2802069246229380770613761295708032929728057828216, 180473260316378408376524889522172090689505315495752, 11623766162470392685815270222093778152221601269182344 ----------------------------- This took, 1340.972, seconds.