This file contains some sample examples of the Gessel-Zeilberger method appl\ ied to Functional Equations of the form A B x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) For various A,B --------------------------------------------------- --------------------------------------------------- [A,B]=, [1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 1 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + --------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 2 3 -1 + 11 x + x + (1 - 14 x + 3 x ) P + x (2 + 3 x) P + x P = 0 and in Maple input notation -1+11*x+x^2+(1-14*x+3*x^2)*P+x*(2+3*x)*P^2+x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590, 58392041019795, 352044769046880, 2132866978427640, 12980019040145352, 79319075627675556, 486556845464525528, 2995168113638767536, 18498288730876090608, 114595331036190018936, 711933341625150895008] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: (3 n + 2) (3 n + 1) A[n] -3/2 ------------------------ + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -3/2*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 8009075383746869367321594089596785085953239124352168812239830730081039132560265\ 3176923402494898230819062859890737376653597051095699941524113699118470496253203\ 3931413493427164058848347525033140661935080120472827911513442094890120839625113\ 3682193246764979474179186494612594181179627296028096563565259355635471707894698\ 880534744 --------------------------------------------------- [A,B]=, [1, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 1 2 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 3 4 -1 + 19 x - 27 x - x + (1 - 28 x + 108 x - 32 x - x ) P 2 3 2 2 3 3 4 - 4 x (-2 + 24 x - 27 x + x ) P - 8 x (-3 + 10 x) P + 16 x (2 + x) P 4 5 + 16 x P = 0 and in Maple input notation -1+19*x-27*x^2-x^3+(1-28*x+108*x^2-32*x^3-x^4)*P-4*x*(-2+24*x-27*x^2+x^3)*P^2-8 *x^2*(-3+10*x)*P^3+16*x^3*(2+x)*P^4+16*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 13, 69, 413, 2679, 18409, 132105, 980601, 7478907, 58322229, 463324077, 3739045685, 30583980591, 253109901713, 2116290212241, 17855500939889, 151867751472051, 1301037847933853, 11218476079117077, 97303919313127373, 848493518872892967, 7435113621177965561, 65444283707074964889, 578423604323891844841, 5131843632817054818987, 45691145318964564897349, 408143613023710171891581, 3656946639671820704873061, 32859500461696305880409439 ] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 9/128 n (3 n + 2) (3 n + 1) (n + 1) 2 (37299511484 n + 261187216159 n + 449312028096) A[n]/((2 n + 11) (2 n + 9) 5 4 (n + 6) (n + 5) %1) + 3/64 (n + 1) (7559717329728 n + 76534478124912 n 3 2 + 276569543854708 n + 429253533713682 n + 265385156754755 n + 36438948082320) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 2 3 9/64 (-4482358503135285 n - 3294632654047901 n - 1194384525274275 n 4 5 6 - 212949375874700 n - 14767893449640 n + 35020022496 n - 2409267128314760) A[n + 2]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 2 3 3/64 (-26280681047290454 n - 15102633079038771 n - 4084239925899070 n 4 5 6 - 462064481358360 n + 277315238304 n + 2790624894336 n - 17732144434876440) A[n + 3]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 4 3 2 3/8 (737235224056 n + 10820090785098 n + 59612339132899 n + 146207851197957 n + 134865050554210) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 13804920856 n + 86406103866 n + 137658848573 and in Maple input format 9/128*n*(3*n+2)*(3*n+1)*(n+1)*(37299511484*n^2+261187216159*n+449312028096)/(2* n+11)/(2*n+9)/(n+6)/(n+5)/(13804920856*n^2+86406103866*n+137658848573)*A[n]+3/ 64*(n+1)*(7559717329728*n^5+76534478124912*n^4+276569543854708*n^3+ 429253533713682*n^2+265385156754755*n+36438948082320)/(2*n+11)/(2*n+9)/(n+6)/(n +5)/(13804920856*n^2+86406103866*n+137658848573)*A[n+1]+9/64*(-4482358503135285 *n-3294632654047901*n^2-1194384525274275*n^3-212949375874700*n^4-14767893449640 *n^5+35020022496*n^6-2409267128314760)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/( 13804920856*n^2+86406103866*n+137658848573)*A[n+2]+3/64*(-26280681047290454*n-\ 15102633079038771*n^2-4084239925899070*n^3-462064481358360*n^4+277315238304*n^5 +2790624894336*n^6-17732144434876440)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(13804920856 *n^2+86406103866*n+137658848573)*A[n+3]-3/8*(737235224056*n^4+10820090785098*n^ 3+59612339132899*n^2+146207851197957*n+134865050554210)/(2*n+11)/(n+6)/( 13804920856*n^2+86406103866*n+137658848573)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 7347349915605799234608879693292900243208361347676741769037736589488285809887794\ 2585643167288016243900143229606228350673949877855278480365815309983806476762334\ 1342066097019602796547620313865943457713507879214194608510845785859881905912868\ 9574136289119940894446805770986935704913144870383402860985277721175265057991897\ 8032168608975167854403298475849219291826737114684254308915199791139942161 --------------------------------------------------- [A,B]=, [1, 3] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 1 3 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 3 4 -1 + 27 x - 87 x - 2 x + (1 - 40 x + 255 x - 106 x - 2 x ) P 2 3 2 2 2 3 - 3 x (-4 + 70 x - 107 x + 5 x ) P - 9 x (-6 + 32 x + 3 x ) P 3 4 4 5 + 27 x (4 + x) P + 81 x P = 0 and in Maple input notation -1+27*x-87*x^2-2*x^3+(1-40*x+255*x^2-106*x^3-2*x^4)*P-3*x*(-4+70*x-107*x^2+5*x^ 3)*P^2-9*x^2*(-6+32*x+3*x^2)*P^3+27*x^3*(4+x)*P^4+81*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [1, 4, 22, 148, 1123, 9232, 80386, 730900, 6873751, 66417796, 656163976, 6603671344, 67511272900, 699549926848, 7333958225482, 77679316326292, 830233081971199, 8945175149670988, 97074771730455862, 1060331516948023924, 11650054338154701643, 128686986060328964416, 1428436825175626066744, 15926907802903223002288, 178316358922204205862436, 2004025508531258876948656, 22601945828095014511236016, 255746750973770013222473632, 2902675745312518468470739816, 33038714321311337084347000672] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 27 --- n (3 n + 2) (3 n + 1) (n + 1) 256 2 (266729362649068 n + 1789592986605783 n + 2848142562033737) A[n]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 1/64 (n + 1) ( 5 4 3 372770155275048904 n + 3653254334353339286 n + 12679911292141675799 n 2 + 18952054670166889696 n + 11298266318378566620 n + 1461261282780913560) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/64 ( 2 528144660619961785830 + 1097129782829429848305 n + 932047246716027386998 n 3 4 + 413628975603898874775 n + 100900146606430876900 n 5 6 + 12797262381581451720 n + 658066379804514592 n ) A[n + 2]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 3/16 (-10740266992223774640 2 3 - 14130395699260845654 n - 5837978792828958981 n - 120503288778349170 n 4 5 6 + 529459248172626640 n + 135506653487907504 n + 10527941989983296 n ) A[n + 3]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/4 ( 4 3 2 11749579673110232 n + 168454986401193866 n + 891816105489304933 n + 2061748281327047644 n + 1753075666642929540) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 116834593206376 n + 652052761296126 n + 898315537501643 and in Maple input format 27/256*n*(3*n+2)*(3*n+1)*(n+1)*(266729362649068*n^2+1789592986605783*n+ 2848142562033737)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n]+1/64*(n+1)*(372770155275048904*n^5+ 3653254334353339286*n^4+12679911292141675799*n^3+18952054670166889696*n^2+ 11298266318378566620*n+1461261282780913560)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/( 116834593206376*n^2+652052761296126*n+898315537501643)*A[n+1]-1/64*( 528144660619961785830+1097129782829429848305*n+932047246716027386998*n^2+ 413628975603898874775*n^3+100900146606430876900*n^4+12797262381581451720*n^5+ 658066379804514592*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n+2]+3/16*(-10740266992223774640-\ 14130395699260845654*n-5837978792828958981*n^2-120503288778349170*n^3+ 529459248172626640*n^4+135506653487907504*n^5+10527941989983296*n^6)/(2*n+11)/( 2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+652052761296126*n+898315537501643)*A[n+ 3]-1/4*(11749579673110232*n^4+168454986401193866*n^3+891816105489304933*n^2+ 2061748281327047644*n+1753075666642929540)/(2*n+11)/(n+6)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 7220432133782328813164045095267903619071719600835429600720473664079446532367674\ 5557509131100778416520153690381826669184976268225736938362533883053821003622179\ 7074265057440254203288945702156531754826232763533303364874280857412056723312715\ 6484685983845174032888275839232550315133020466077311718262165326822891583283030\ 4312606882073685392002681646526120096836299071909070662507208846702259452983955\ 25443779156057077668100007983212432 --------------------------------------------------- [A,B]=, [2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 2 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + --------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 3 4 -2 - 54 x - 2 x + 38 x + (1 - 28 x + 108 x - 32 x - x ) P 2 3 2 2 3 3 4 - 2 x (-2 + 24 x - 27 x + x ) P - 2 x (-3 + 10 x) P + 2 x (2 + x) P 4 5 + x P = 0 and in Maple input notation -2-54*x^2-2*x^3+38*x+(1-28*x+108*x^2-32*x^3-x^4)*P-2*x*(-2+24*x-27*x^2+x^3)*P^2 -2*x^2*(-3+10*x)*P^3+2*x^3*(2+x)*P^4+x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [2, 6, 26, 138, 826, 5358, 36818, 264210, 1961202, 14957814, 116644458, 926648154, 7478091370, 61167961182, 506219803426, 4232580424482, 35711001879778 , 303735502944102, 2602075695867706, 22436952158234154, 194607838626254746, 1696987037745785934, 14870227242355931122, 130888567414149929778, 1156847208647783689682, 10263687265634109637974, 91382290637929129794698, 816287226047420343783162, 7313893279343641409746122, 65719000923392611760818878 ] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 9/128 n (3 n + 2) (3 n + 1) (n + 1) 2 (37299511484 n + 261187216159 n + 449312028096) A[n]/((2 n + 11) (2 n + 9) 5 4 (n + 6) (n + 5) %1) + 3/64 (n + 1) (7559717329728 n + 76534478124912 n 3 2 + 276569543854708 n + 429253533713682 n + 265385156754755 n + 36438948082320) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 2 3 9/64 (-4482358503135285 n - 3294632654047901 n - 1194384525274275 n 4 5 6 - 212949375874700 n - 14767893449640 n + 35020022496 n - 2409267128314760) A[n + 2]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 2 3 3/64 (-26280681047290454 n - 15102633079038771 n - 4084239925899070 n 4 5 6 - 462064481358360 n + 277315238304 n + 2790624894336 n - 17732144434876440) A[n + 3]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 4 3 2 3/8 (737235224056 n + 10820090785098 n + 59612339132899 n + 146207851197957 n + 134865050554210) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 13804920856 n + 86406103866 n + 137658848573 and in Maple input format 9/128*n*(3*n+2)*(3*n+1)*(n+1)*(37299511484*n^2+261187216159*n+449312028096)/(2* n+11)/(2*n+9)/(n+6)/(n+5)/(13804920856*n^2+86406103866*n+137658848573)*A[n]+3/ 64*(n+1)*(7559717329728*n^5+76534478124912*n^4+276569543854708*n^3+ 429253533713682*n^2+265385156754755*n+36438948082320)/(2*n+11)/(2*n+9)/(n+6)/(n +5)/(13804920856*n^2+86406103866*n+137658848573)*A[n+1]+9/64*(-4482358503135285 *n-3294632654047901*n^2-1194384525274275*n^3-212949375874700*n^4-14767893449640 *n^5+35020022496*n^6-2409267128314760)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/( 13804920856*n^2+86406103866*n+137658848573)*A[n+2]+3/64*(-26280681047290454*n-\ 15102633079038771*n^2-4084239925899070*n^3-462064481358360*n^4+277315238304*n^5 +2790624894336*n^6-17732144434876440)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(13804920856 *n^2+86406103866*n+137658848573)*A[n+3]-3/8*(737235224056*n^4+10820090785098*n^ 3+59612339132899*n^2+146207851197957*n+134865050554210)/(2*n+11)/(n+6)/( 13804920856*n^2+86406103866*n+137658848573)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 1469469983121159846921775938658580048641672269535348353807547317897657161977558\ 8517128633457603248780028645921245670134789975571055696073163061996761295352466\ 8268413219403920559309524062773188691542701575842838921702169157171976381182573\ 7914827257823988178889361154197387140982628974076680572197055544235053011598379\ 56064337217950335708806596951698438583653474229368508617830399582279884322 --------------------------------------------------- [A,B]=, [2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 2 2 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 3 4 2 70 x - 358 x - 6 x - 2 + (-220 x - 3 x + 1 - 52 x + 466 x ) P 2 3 2 2 2 3 3 4 - 8 x (23 x - 44 x + 2 x - 1) P - 8 x (-3 + 22 x + 3 x ) P + 32 x P 4 5 + 16 x P = 0 and in Maple input notation 70*x-358*x^2-6*x^3-2+(-220*x^3-3*x^4+1-52*x+466*x^2)*P-8*x*(23*x-44*x^2+2*x^3-1 )*P^2-8*x^2*(-3+22*x+3*x^2)*P^3+32*x^3*P^4+16*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [2, 10, 66, 530, 4802, 47130, 489858, 5316130, 59669634, 688094250, 8112723906, 97436323890, 1188736049474, 14699269169210, 183898728138498, 2324369744016450, 29645331374818562, 381152725261367370, 4935922408801776450, 64335927643101644370, 843506137799145737154, 11118392152573396899930, 147270003814116005316738, 1959429377092092418335330, 26177811439054189885290882 , 351065865891163786226586730, 4724694195509261324647559874, 63794008517966650877913547890, 863991811181341772264494969410, 11734783341065722797113547194490] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 3/2 n (3 n + 2) (3 n + 1) (n + 1) 2 (1789515304126 n + 11545556936517 n + 17129666734370) A[n]/((2 n + 11) 5 (2 n + 9) (n + 6) (n + 5) %1) + 1/4 (n + 1) (3129753488112892 n 4 3 2 + 30256540961200865 n + 104246809195390805 n + 158841157202864170 n + 103144606571811933 n + 19869561465636330) A[n + 1]/((2 n + 11) (2 n + 9) 2 (n + 6) (n + 5) %1) - 5/8 (3338510628018542868 n + 2910482223783373475 n 3 4 5 + 1329964929521525646 n + 335624321956495904 n + 44318859807746736 n 6 + 2392805287872128 n + 1569963802803962580) A[n + 2]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 15/8 (340950062972163301 n 2 3 4 + 278465694056743172 n + 121487224141274054 n + 29815949588317376 n 5 6 + 3894787105548240 n + 211094710710848 n + 174211806537067374) A[n + 3]/ 4 ((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/8 (1505750458433792 n 3 2 + 21390909615924464 n + 111807089011285108 n + 253974941119408711 n + 210805697785941210) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 6035358202924 n + 32674653547125 n + 43241530566821 and in Maple input format 3/2*n*(3*n+2)*(3*n+1)*(n+1)*(1789515304126*n^2+11545556936517*n+17129666734370) /(2*n+11)/(2*n+9)/(n+6)/(n+5)/(6035358202924*n^2+32674653547125*n+ 43241530566821)*A[n]+1/4*(n+1)*(3129753488112892*n^5+30256540961200865*n^4+ 104246809195390805*n^3+158841157202864170*n^2+103144606571811933*n+ 19869561465636330)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(6035358202924*n^2+ 32674653547125*n+43241530566821)*A[n+1]-5/8*(3338510628018542868*n+ 2910482223783373475*n^2+1329964929521525646*n^3+335624321956495904*n^4+ 44318859807746736*n^5+2392805287872128*n^6+1569963802803962580)/(2*n+11)/(2*n+9 )/(n+6)/(n+5)/(6035358202924*n^2+32674653547125*n+43241530566821)*A[n+2]+15/8*( 340950062972163301*n+278465694056743172*n^2+121487224141274054*n^3+ 29815949588317376*n^4+3894787105548240*n^5+211094710710848*n^6+ 174211806537067374)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(6035358202924*n^2+ 32674653547125*n+43241530566821)*A[n+3]-1/8*(1505750458433792*n^4+ 21390909615924464*n^3+111807089011285108*n^2+253974941119408711*n+ 210805697785941210)/(2*n+11)/(n+6)/(6035358202924*n^2+32674653547125*n+ 43241530566821)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 6320398307664129281258965046129315380695412196515991226174271537311607898486713\ 7088463781460683624927158387320280704935963140645448189038297450538699520474253\ 7887506727515250454984302812549207982712136201336848162219960109071347145339939\ 5346700537996131377856005792712721043804242247440648308970581591997642808050917\ 4188165112537129699040345481565494189659537515871167463066331489526764960268436\ 293163711396254102009031363105284937156244766998896554191012534850 --------------------------------------------------- [A,B]=, [2, 3] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 2 3 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 3 4 2 -2 + 102 x - 918 x - 10 x + (-568 x - 5 x - 76 x + 1080 x + 1) P 2 3 2 2 2 3 - 6 x (68 x - 181 x + 7 x - 2) P - 18 x (-3 + 34 x + 6 x ) P 3 4 4 5 - 54 x (-2 + x) P + 81 x P = 0 and in Maple input notation -2+102*x-918*x^2-10*x^3+(-568*x^3-5*x^4-76*x+1080*x^2+1)*P-6*x*(68*x-181*x^2+7* x^3-2)*P^2-18*x^2*(-3+34*x+6*x^2)*P^3-54*x^3*(-2+x)*P^4+81*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [2, 14, 122, 1274, 15002, 191366, 2584466, 36437618, 531265970, 7957508510, 121854651530, 1900747423274, 30116410379402, 483634141882262, 7857687701118626, 128976064487141858, 2136202297391212130, 35666666497495241390, 599798343831398102810, 10152209765154655581530, 172847244106548237984890, 2958563646368405441628710, 50888047845793142861863730, 879208327485118968157514834, 15252974625692727455170264082, 265624050741888237760843993406, 4642038937393460753995868034602, 81389822016973263879865445393546, 1431376665562933810734446770425962, 25244836072791624765049336962697526] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 189 --- n (3 n + 2) (3 n + 1) (n + 1) 64 2 (24200777443099316 n + 149981599812317901 n + 206282348197876024) A[n]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 1/32 (n + 1) ( 5 4 1064821178254625432392 n + 10179838529153752802786 n 3 2 + 34966137898222241553494 n + 54552574522847928514066 n + 38375287012365525102087 n + 9397426882254128798040) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 7/32 (133250551273255132566720 2 + 288218531774582815451193 n + 255956395605881335023811 n 3 4 + 119400310337914147225905 n + 30851298770038265454820 n 5 6 + 4187375063301767645592 n + 233437175788782948064 n ) A[n + 2]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 3/32 (174111106166728525993680 2 + 324064002078172329292362 n + 249413367998023268293611 n 3 4 + 101827053251900792844690 n + 23289540215508253466920 n 5 6 + 2831348035438659667968 n + 142976745068494671104 n ) A[n + 3]/( 4 (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 7/8 (2679506209060469648 n 3 2 + 37738801973922884012 n + 195103805608414002526 n + 436910749174046708713 n + 355841454074943516126) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 53806314568501508 n + 284427914721874329 n + 365079009424202323 and in Maple input format 189/64*n*(3*n+2)*(3*n+1)*(n+1)*(24200777443099316*n^2+149981599812317901*n+ 206282348197876024)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508*n^2+ 284427914721874329*n+365079009424202323)*A[n]+1/32*(n+1)*( 1064821178254625432392*n^5+10179838529153752802786*n^4+34966137898222241553494* n^3+54552574522847928514066*n^2+38375287012365525102087*n+ 9397426882254128798040)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508*n^2+ 284427914721874329*n+365079009424202323)*A[n+1]-7/32*(133250551273255132566720+ 288218531774582815451193*n+255956395605881335023811*n^2+ 119400310337914147225905*n^3+30851298770038265454820*n^4+4187375063301767645592 *n^5+233437175788782948064*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508 *n^2+284427914721874329*n+365079009424202323)*A[n+2]+3/32*( 174111106166728525993680+324064002078172329292362*n+249413367998023268293611*n^ 2+101827053251900792844690*n^3+23289540215508253466920*n^4+ 2831348035438659667968*n^5+142976745068494671104*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n +5)/(53806314568501508*n^2+284427914721874329*n+365079009424202323)*A[n+3]-7/8* (2679506209060469648*n^4+37738801973922884012*n^3+195103805608414002526*n^2+ 436910749174046708713*n+355841454074943516126)/(2*n+11)/(n+6)/( 53806314568501508*n^2+284427914721874329*n+365079009424202323)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 1278545232298582531064080455207659734305065429857930328975114935541451318037808\ 0335857513683144289176458697036810691655074782003935097064075423578637539835730\ 8113956169769092864095179766611665159423347936047225691823848161873201283978157\ 7887898846083325088345920599599168190913263178529656398380879711875662808396973\ 0214448258746241382159436364836091190514978637801752440678611722986061343690362\ 1121260003158206028067267701919777329115648082906120703585195591625775212491240\ 488208271314531484068512463253218 --------------------------------------------------- [A,B]=, [3, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 3 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + --------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 3 4 81 x - 261 x - 6 x - 3 + (1 - 40 x + 255 x - 106 x - 2 x ) P 2 3 2 2 2 3 - x (-4 + 70 x - 107 x + 5 x ) P - x (-6 + 32 x + 3 x ) P 3 4 4 5 + x (4 + x) P + x P = 0 and in Maple input notation 81*x-261*x^2-6*x^3-3+(1-40*x+255*x^2-106*x^3-2*x^4)*P-x*(-4+70*x-107*x^2+5*x^3) *P^2-x^2*(-6+32*x+3*x^2)*P^3+x^3*(4+x)*P^4+x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [3, 12, 66, 444, 3369, 27696, 241158, 2192700, 20621253, 199253388, 1968491928, 19811014032, 202533818700, 2098649780544, 22001874676446, 233037948978876, 2490699245913597, 26835525449012964, 291224315191367586, 3180994550844071772, 34950163014464104929, 386060958180986893248, 4285310475526878200232, 47780723408709669006864, 534949076766612617587308, 6012076525593776630845968, 67805837484285043533708048, 767240252921310039667420896, 8708027235937555405412219448, 99116142963934011253041002016] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 27 --- n (3 n + 2) (3 n + 1) (n + 1) 256 2 (266729362649068 n + 1789592986605783 n + 2848142562033737) A[n]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 1/64 (n + 1) ( 5 4 3 372770155275048904 n + 3653254334353339286 n + 12679911292141675799 n 2 + 18952054670166889696 n + 11298266318378566620 n + 1461261282780913560) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/64 ( 2 528144660619961785830 + 1097129782829429848305 n + 932047246716027386998 n 3 4 + 413628975603898874775 n + 100900146606430876900 n 5 6 + 12797262381581451720 n + 658066379804514592 n ) A[n + 2]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 3/16 (-10740266992223774640 2 3 - 14130395699260845654 n - 5837978792828958981 n - 120503288778349170 n 4 5 6 + 529459248172626640 n + 135506653487907504 n + 10527941989983296 n ) A[n + 3]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/4 ( 4 3 2 11749579673110232 n + 168454986401193866 n + 891816105489304933 n + 2061748281327047644 n + 1753075666642929540) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 116834593206376 n + 652052761296126 n + 898315537501643 and in Maple input format 27/256*n*(3*n+2)*(3*n+1)*(n+1)*(266729362649068*n^2+1789592986605783*n+ 2848142562033737)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n]+1/64*(n+1)*(372770155275048904*n^5+ 3653254334353339286*n^4+12679911292141675799*n^3+18952054670166889696*n^2+ 11298266318378566620*n+1461261282780913560)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/( 116834593206376*n^2+652052761296126*n+898315537501643)*A[n+1]-1/64*( 528144660619961785830+1097129782829429848305*n+932047246716027386998*n^2+ 413628975603898874775*n^3+100900146606430876900*n^4+12797262381581451720*n^5+ 658066379804514592*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n+2]+3/16*(-10740266992223774640-\ 14130395699260845654*n-5837978792828958981*n^2-120503288778349170*n^3+ 529459248172626640*n^4+135506653487907504*n^5+10527941989983296*n^6)/(2*n+11)/( 2*n+9)/(n+6)/(n+5)/(116834593206376*n^2+652052761296126*n+898315537501643)*A[n+ 3]-1/4*(11749579673110232*n^4+168454986401193866*n^3+891816105489304933*n^2+ 2061748281327047644*n+1753075666642929540)/(2*n+11)/(n+6)/(116834593206376*n^2+ 652052761296126*n+898315537501643)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 2166129640134698643949213528580371085721515880250628880216142099223833959710302\ 3667252739330233524956046107114548000755492880467721081508760164916146301086653\ 9122279517232076260986683710646959526447869829059991009462284257223617016993814\ 6945405795153552209866482751769765094539906139823193515478649598046867474984909\ 1293782064622105617600804493957836029050889721572721198752162654010677835895186\ 576331337468171233004300023949637296 --------------------------------------------------- [A,B]=, [3, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 3 2 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 3 4 2 153 x - 1377 x - 15 x - 3 + (-568 x - 5 x - 76 x + 1080 x + 1) P 2 3 2 2 2 3 - 4 x (68 x - 181 x + 7 x - 2) P - 8 x (-3 + 34 x + 6 x ) P 3 4 4 5 - 16 x (-2 + x) P + 16 x P = 0 and in Maple input notation 153*x-1377*x^2-15*x^3-3+(-568*x^3-5*x^4-76*x+1080*x^2+1)*P-4*x*(68*x-181*x^2+7* x^3-2)*P^2-8*x^2*(-3+34*x+6*x^2)*P^3-16*x^3*(-2+x)*P^4+16*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [3, 21, 183, 1911, 22503, 287049, 3876699, 54656427, 796898955, 11936262765, 182781977295, 2851121134911, 45174615569103, 725451212823393, 11786531551677939 , 193464096730712787, 3204303446086818195, 53499999746242862085, 899697515747097154215, 15228314647731983372295, 259270866159822356977335, 4437845469552608162443065, 76332071768689714292795595, 1318812491227678452236272251, 22879461938539091182755396123, 398436076112832356641265990109, 6963058406090191130993802051903, 122084733025459895819798168090319, 2147064998344400716101670155638943, 37867254109187437147574005444046289] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 189 --- n (3 n + 2) (3 n + 1) (n + 1) 64 2 (24200777443099316 n + 149981599812317901 n + 206282348197876024) A[n]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 1/32 (n + 1) ( 5 4 1064821178254625432392 n + 10179838529153752802786 n 3 2 + 34966137898222241553494 n + 54552574522847928514066 n + 38375287012365525102087 n + 9397426882254128798040) A[n + 1]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 7/32 (133250551273255132566720 2 + 288218531774582815451193 n + 255956395605881335023811 n 3 4 + 119400310337914147225905 n + 30851298770038265454820 n 5 6 + 4187375063301767645592 n + 233437175788782948064 n ) A[n + 2]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 3/32 (174111106166728525993680 2 + 324064002078172329292362 n + 249413367998023268293611 n 3 4 + 101827053251900792844690 n + 23289540215508253466920 n 5 6 + 2831348035438659667968 n + 142976745068494671104 n ) A[n + 3]/( 4 (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 7/8 (2679506209060469648 n 3 2 + 37738801973922884012 n + 195103805608414002526 n + 436910749174046708713 n + 355841454074943516126) A[n + 4]/((2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 53806314568501508 n + 284427914721874329 n + 365079009424202323 and in Maple input format 189/64*n*(3*n+2)*(3*n+1)*(n+1)*(24200777443099316*n^2+149981599812317901*n+ 206282348197876024)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508*n^2+ 284427914721874329*n+365079009424202323)*A[n]+1/32*(n+1)*( 1064821178254625432392*n^5+10179838529153752802786*n^4+34966137898222241553494* n^3+54552574522847928514066*n^2+38375287012365525102087*n+ 9397426882254128798040)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508*n^2+ 284427914721874329*n+365079009424202323)*A[n+1]-7/32*(133250551273255132566720+ 288218531774582815451193*n+255956395605881335023811*n^2+ 119400310337914147225905*n^3+30851298770038265454820*n^4+4187375063301767645592 *n^5+233437175788782948064*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(53806314568501508 *n^2+284427914721874329*n+365079009424202323)*A[n+2]+3/32*( 174111106166728525993680+324064002078172329292362*n+249413367998023268293611*n^ 2+101827053251900792844690*n^3+23289540215508253466920*n^4+ 2831348035438659667968*n^5+142976745068494671104*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n +5)/(53806314568501508*n^2+284427914721874329*n+365079009424202323)*A[n+3]-7/8* (2679506209060469648*n^4+37738801973922884012*n^3+195103805608414002526*n^2+ 436910749174046708713*n+355841454074943516126)/(2*n+11)/(n+6)/( 53806314568501508*n^2+284427914721874329*n+365079009424202323)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 1917817848447873796596120682811489601457598144786895493462672403312176977056712\ 0503786270524716433764688045555216037482612173005902645596113135367956309753596\ 2170934254653639296142769649917497739135021904070838537735772242809801925967236\ 6831848269124987632518880899398752286369894767794484597571319567813494212595459\ 5321672388119362073239154547254136785772467956702628661017917584479092015535543\ 1681890004737309042100901552879665993673472124359181055377793387438662818736860\ 732312406971797226102768694879827 --------------------------------------------------- [A,B]=, [3, 3] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 3 3 x y (f(x, 1) - y f(x, y)) (f(x, 1) - f(x, y)) f(x, y) = ------- + ----------------------------------------------- 1 - x y 2 (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 3 4 225 x - 3357 x - 24 x - 3 + (1 - 112 x + 2481 x - 1390 x - 8 x ) P 2 3 2 2 2 3 - 3 x (-4 + 202 x - 761 x + 23 x ) P - 9 x (-6 + 104 x + 21 x ) P 3 4 4 5 - 27 x (-4 + 5 x) P + 81 x P = 0 and in Maple input notation 225*x-3357*x^2-24*x^3-3+(1-112*x+2481*x^2-1390*x^3-8*x^4)*P-3*x*(-4+202*x-761*x ^2+23*x^3)*P^2-9*x^2*(-6+104*x+21*x^2)*P^3-27*x^3*(-4+5*x)*P^4+81*x^4*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [3, 30, 354, 4890, 75873, 1275240, 22690182, 421372290, 8091117381, 159591042750, 3217927551384, 66090258856920, 1378722646670556, 29149901047962480, 623519462194803102, 13473807797456360850, 293793438941754850173, 6457650125014636929210, 142963218749385810163650, 3185534333750592559282410, 71397507400147094463348681, 1608781800482167649803451040, 36427120488090308211473359272, 828497517362192469493972034040, 18920916851105863207674118652268, 433751842347688749528463424624040, 9978547648503012312151383469047696, 230309456285864029303030781793663120, 5331843242658997915581379695995633640, 123787431831136871118202586105152660320] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 243 --- n (3 n + 2) (3 n + 1) (n + 1) 128 2 (5255943897125055604 n + 31834903725485174073 n + 41851702676941496705) A[n]/((2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + 1/64 (n + 1) ( 5 4 463368834846544213133684 n + 4399861836081948601790365 n 3 2 + 15068671598547375132895045 n + 23732122821727629993836900 n + 17243264160012234307969476 n + 4571730709905595500338760) A[n + 1]/( (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 5/32 (26906429888343920712930930 2 + 58752061096720090039556283 n + 52722585984205132139247836 n 3 4 + 24886613698445100341650959 n + 6518802011867756098799636 n 5 6 + 899000215422786433617768 n + 51048369294084446377376 n ) A[n + 2]/( 15 (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) + -- (2333900408180977444213296 16 2 + 4328256143440565453516334 n + 3308417720733477546906243 n 3 4 + 1337223266048803288485141 n + 301903660713317758779364 n 5 6 + 36137794531221249885240 n + 1793097139014608556512 n ) A[n + 3]/( 4 (2 n + 11) (2 n + 9) (n + 6) (n + 5) %1) - 1/4 (574207343790502120336 n 3 2 + 8043593875195189968412 n + 41313090820390081976234 n + 91762646985672819212663 n + 73953174344457068291010) A[n + 4]/( (2 n + 11) (n + 6) %1) + A[n + 5] = 0 2 %1 := 2304931534175007908 n + 12021011281266590145 n + 15175524774382496977 and in Maple input format 243/128*n*(3*n+2)*(3*n+1)*(n+1)*(5255943897125055604*n^2+31834903725485174073*n +41851702676941496705)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(2304931534175007908*n^2+ 12021011281266590145*n+15175524774382496977)*A[n]+1/64*(n+1)*( 463368834846544213133684*n^5+4399861836081948601790365*n^4+ 15068671598547375132895045*n^3+23732122821727629993836900*n^2+ 17243264160012234307969476*n+4571730709905595500338760)/(2*n+11)/(2*n+9)/(n+6)/ (n+5)/(2304931534175007908*n^2+12021011281266590145*n+15175524774382496977)*A[n +1]-5/32*(26906429888343920712930930+58752061096720090039556283*n+ 52722585984205132139247836*n^2+24886613698445100341650959*n^3+ 6518802011867756098799636*n^4+899000215422786433617768*n^5+ 51048369294084446377376*n^6)/(2*n+11)/(2*n+9)/(n+6)/(n+5)/(2304931534175007908* n^2+12021011281266590145*n+15175524774382496977)*A[n+2]+15/16*( 2333900408180977444213296+4328256143440565453516334*n+3308417720733477546906243 *n^2+1337223266048803288485141*n^3+301903660713317758779364*n^4+ 36137794531221249885240*n^5+1793097139014608556512*n^6)/(2*n+11)/(2*n+9)/(n+6)/ (n+5)/(2304931534175007908*n^2+12021011281266590145*n+15175524774382496977)*A[n +3]-1/4*(574207343790502120336*n^4+8043593875195189968412*n^3+ 41313090820390081976234*n^2+91762646985672819212663*n+73953174344457068291010)/ (2*n+11)/(n+6)/(2304931534175007908*n^2+12021011281266590145*n+ 15175524774382496977)*A[n+4]+A[n+5] = 0 Just for fun, A[400], equals 9168551283779344238254913830091548363508172536185081487799224787553965525810007\ 7410467616550177716636016122647202548929638244197519177835927572294051246166810\ 8456976831402895573773582248194772772319276831818416763725791249652936653366154\ 6082679372377800378120499470628781047125908069175202425147809421547824353816385\ 7308488269053379135149161773411875264247376181730973668128053303706604171546426\ 6208634807078854285674391664152230462351212465071421065032690950785444663156177\ 9447764508724079644664292515710773327503729425378476413934005590529600848882980\ 0 --------------------------------------------------- This ends this file, that took, 477.589, seconds.