This file contains some sample examples of the Gessel-Zeilberger method appl\ ied to Functional Equations of the form / a (b + 1) \ | a C (y f(x, 1) - y f(x, y))| f(x, y) = x (A + f(x, y)) |B y + ---------------------------------| \ 1 - y / For various a,b,A,B,C --------------------------------------------------- Let's take a=0, b=1 --------------------------------------------------- [A,B,C]=, [1, 1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | f(x, 1) - y f(x, y)| f(x, y) = x (1 + f(x, y)) |1 + --------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 x (-1 + 9 x) + (1 - 12 x + 24 x ) P + 16 x P = 0 and in Maple input notation x*(-1+9*x)+(1-12*x+24*x^2)*P+16*x^2*P^2 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, 2149888, 13891584, 91287552, 608583680, 4107939840, 28030648320, 193100021760, 1341536993280, 9390758952960, 66182491668480, 469294031831040, 3346270487838720, 23981605162844160, 172667557172477952, 1248519259554840576, 9063324995286990848, 66032796394233790464, 482722511571640123392, 3539965084858694238208, 26035872237025235042304, 192014557748061108436992] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 4 (1 + 2 n) A[n] - ---------------- + A[n + 1] = 0 3 + n and in Maple input format -4*(1+2*n)/(3+n)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 4518299145696574400748649707519785060334188410016601616805788337756023095244134\ 2746520670185143041063327119403562851049480330085453473995782490176797106787883\ 0619018845113790848059829100359704192765525842321346649290918884611118891307134\ 4523113849135234730869455488783609958845617409371336080514467328151492075659393\ 834398390005840472687936639995457044480 --------------------------------------------------- [A,B,C]=, [1, 1, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | 2 (f(x, 1) - y f(x, y))| f(x, y) = x (1 + f(x, y)) |1 + ------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 x (1 - 18 x + 49 x ) + (-1 + 23 x - 135 x + 273 x ) P 2 2 3 3 + 4 x (1 - 36 x + 120 x ) P + 256 x P = 0 and in Maple input notation x*(1-18*x+49*x^2)+(-1+23*x-135*x^2+273*x^3)*P+4*x*(1-36*x+120*x^2)*P^2+256*x^3* P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, 5, 33, 253, 2133, 19209, 181597, 1781993, 18010905, 186460925, 1969136073, 21146402949, 230359997325, 2540633027361, 28324195371045, 318781746722449, 3618129551647473, 41375084451654837, 476350498939657521, 5517777122957208525, 64269878549045224581, 752393163510060403449, 8848940589636911399853, 104516363993092095973113, 1239311100247632301005193, 14748640803628320646901549 , 176110658447007194048739417, 2109499888556841327788730581, 25342071328575550049199410493, 305274756452915383966257216657] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 108 n (3 n + 7) (1 + 2 n) A[n] 3 (3 n + 5) (27 n + 99 n + 44) A[n + 1] - ------------------------------ + ---------------------------------------- (n + 5) (n + 4) (3 n + 4) (n + 5) (n + 4) (3 n + 4) 2 3 (18 n + 69 n + 55) A[2 + n] - ------------------------------ + A[3 + n] = 0 (n + 5) (3 n + 4) and in Maple input format -108*n*(3*n+7)*(1+2*n)/(n+5)/(n+4)/(3*n+4)*A[n]+3*(3*n+5)*(27*n^2+99*n+44)/(n+5 )/(n+4)/(3*n+4)*A[n+1]-3*(18*n^2+69*n+55)/(n+5)/(3*n+4)*A[2+n]+A[3+n] = 0 Just for fun, A[400], equals 4848760182846786262128855184943642147327576709279073395302311276348943950026610\ 2812934689599335879063395382870625739280529289268672221990549007352498412884081\ 8760763899717209801474072161463054820240061034851038876676285430069248265722456\ 1371693998882794030728159808049762220105389204684268077046989252513869493239078\ 7980380788934686715780539688267016409116764693530128413140019774070513071341629\ 624196337528035768390136322585196795960391697 --------------------------------------------------- [A,B,C]=, [1, 2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | f(x, 1) - y f(x, y)| f(x, y) = x (1 + f(x, y)) |2 + --------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 2 2 4 x (1 - 12 x + 4 x ) + (-2 + 32 x - 96 x + 48 x ) P + x (1 - 48 x + 48 x ) P 3 3 + 16 x P = 0 and in Maple input notation 4*x*(1-12*x+4*x^2)+(-2+32*x-96*x^2+48*x^3)*P+x*(1-48*x+48*x^2)*P^2+16*x^3*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [2, 8, 42, 256, 1716, 12288, 92378, 720896, 5794620, 47710208, 400720260, 3422552064, 29653299240, 260113956864, 2306416171770, 20645907791872, 186374612055660, 1695137692385280, 15522342689778540, 143007979866685440, 1324859557482520920, 12335992698065387520, 115395267456727572132, 1084049081405376823296, 10223823439798615493336, 96772889425775226257408, 919086472674310020383688, 8756277994939110127566848, 83666516964093641210411856 , 801623127407651436839829504] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 12 (3 n + 4) (3 n + 2) A[n] - --------------------------- + A[2 + n] = 0 (n + 4) (3 + n) and in Maple input format -12*(3*n+4)*(3*n+2)/(n+4)/(3+n)*A[n]+A[2+n] = 0 Just for fun, A[400], equals 2073195462534446447630947658301108953463837965069437998574242475216160723193854\ 1978092370953448014698757407469765423880496191989333461368727827495726270200191\ 6014444661626084087822131251901067985373214887907344362705339890113832260460631\ 9990584270894536846997309888958743076212469899363482711275965114001389471596527\ 3982757387681230046637786795543861816708502541923169332814661850304882041270090\ 858496 --------------------------------------------------- [A,B,C]=, [1, 2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | 2 (f(x, 1) - y f(x, y))| f(x, y) = x (1 + f(x, y)) |2 + ------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 2 x (-1 + 18 x) + (1 - 24 x + 96 x ) P + 64 x P = 0 and in Maple input notation 2*x*(-1+18*x)+(1-24*x+96*x^2)*P+64*x^2*P^2 = 0 The first, 30, terms, for the sake of the OEIS are [2, 12, 96, 896, 9216, 101376, 1171456, 14057472, 173801472, 2201485312, 28449964032, 373913812992, 4985517506560, 67304486338560, 918508284149760, 12655003026063360, 175837936783196160, 2461731114964746240, 34698686191884042240, 492090458721264599040, 7017637846111947325440, 100586142460937911664640, 1448440451437505927970816, 20946677297711624189116416 , 304114722248257654893838336, 4431385952760325828453072896, 64789918757599246595313893376, 950252141778122283397937102848, 13977902472607218104176107061248, 206174061470956467036597579153408] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 8 (1 + 2 n) A[n] - ---------------- + A[n + 1] = 0 3 + n and in Maple input format -8*(1+2*n)/(3+n)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 1166737741813516247774715755371749933775629349284390005922764286498386217558684\ 1491834835007166522430146939297080303893586746197575764853207833513918943952092\ 0286167754921270803498656038970044942430113568935046565065281907039416003268341\ 3748580458301630791100699813700731501429435190496635886077909085290103253281098\ 2108431474843001308890036785801404403458621621816822929835025136312136567265656\ 9734178772761627500874930869394227402821696633323627242297922085941254013373644\ 80 --------------------------------------------------- [A,B,C]=, [2, 1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | f(x, 1) - y f(x, y)| f(x, y) = x (2 + f(x, y)) |1 + --------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 2 x (1 - 18 x + 49 x ) + (-1 + 23 x - 135 x + 273 x ) P 2 2 3 3 + 2 x (1 - 36 x + 120 x ) P + 64 x P = 0 and in Maple input notation 2*x*(1-18*x+49*x^2)+(-1+23*x-135*x^2+273*x^3)*P+2*x*(1-36*x+120*x^2)*P^2+64*x^3 *P^3 = 0 The first, 30, terms, for the sake of the OEIS are [2, 10, 66, 506, 4266, 38418, 363194, 3563986, 36021810, 372921850, 3938272146, 42292805898, 460719994650, 5081266054722, 56648390742090, 637563493444898, 7236259103294946, 82750168903309674, 952700997879315042, 11035554245914417050, 128539757098090449162, 1504786327020120806898, 17697881179273822799706, 209032727986184191946226, 2478622200495264602010386, 29497281607256641293803098 , 352221316894014388097478834, 4218999777113682655577461162, 50684142657151100098398820986, 610549512905830767932514433314] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 108 n (3 n + 7) (1 + 2 n) A[n] 3 (3 n + 5) (27 n + 99 n + 44) A[n + 1] - ------------------------------ + ---------------------------------------- (n + 5) (n + 4) (3 n + 4) (n + 5) (n + 4) (3 n + 4) 2 3 (18 n + 69 n + 55) A[n + 2] - ------------------------------ + A[3 + n] = 0 (n + 5) (3 n + 4) and in Maple input format -108*n*(3*n+7)*(1+2*n)/(n+5)/(n+4)/(3*n+4)*A[n]+3*(3*n+5)*(27*n^2+99*n+44)/(n+5 )/(n+4)/(3*n+4)*A[n+1]-3*(18*n^2+69*n+55)/(n+5)/(3*n+4)*A[n+2]+A[3+n] = 0 Just for fun, A[400], equals 9697520365693572524257710369887284294655153418558146790604622552697887900053220\ 5625869379198671758126790765741251478561058578537344443981098014704996825768163\ 7521527799434419602948144322926109640480122069702077753352570860138496531444912\ 2743387997765588061456319616099524440210778409368536154093978505027738986478157\ 5960761577869373431561079376534032818233529387060256826280039548141026142683259\ 248392675056071536780272645170393591920783394 --------------------------------------------------- [A,B,C]=, [2, 1, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | 2 (f(x, 1) - y f(x, y))| f(x, y) = x (2 + f(x, y)) |1 + ------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 2 x (25 x - 1) (-1 + 9 x) + (-1 + 43 x - 523 x + 2145 x ) P 2 2 3 3 + 8 x (1 - 52 x + 376 x ) P + 1024 x P = 0 and in Maple input notation 2*x*(25*x-1)*(-1+9*x)+(-1+43*x-523*x^2+2145*x^3)*P+8*x*(1-52*x+376*x^2)*P^2+ 1024*x^3*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [2, 18, 210, 2818, 41346, 645714, 10562194, 179062146, 3123413634, 55763694098, 1015006961106, 18779256431490, 352336956002050, 6691030243617618, 128416655078209170, 2487707701316942082, 48593061486394249986, 956236356726479866386, 18943006928029519491282, 377527240040322013854210, 7565268777832954438564482, 152359462816093126819443282, 3082491526014775965119064210, 62627058418630807431356430978, 1277346088293375887787217326466, 26146655011171549119320920958994, 536998052806190645578336916216274, 11063128763868188017305513115201154, 228581795213332424624819050789700610, 4735678689426026535292219441572335442] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 1372 n (3 n + 7) (1 + 2 n) A[n] (3 n + 5) (561 n + 2057 n + 1092) A[n + 1] - ------------------------------- + ------------------------------------------- (n + 5) (n + 4) (3 n + 4) (n + 5) (n + 4) (3 n + 4) 2 7 (18 n + 69 n + 55) A[n + 2] - ------------------------------ + A[3 + n] = 0 (n + 5) (3 n + 4) and in Maple input format -1372*n*(3*n+7)*(1+2*n)/(n+5)/(n+4)/(3*n+4)*A[n]+(3*n+5)*(561*n^2+2057*n+1092)/ (n+5)/(n+4)/(3*n+4)*A[n+1]-7*(18*n^2+69*n+55)/(n+5)/(3*n+4)*A[n+2]+A[3+n] = 0 Just for fun, A[400], equals 9222187622237828229521639565524215645228049530138029318980979498512497383532231\ 2158765476313505563025043107490305590552274513153594020134166279934897222230239\ 3715831185168212424802183352999637854086089196175099826245908386796522401255625\ 3962679394743920461441007007435892455428016114479690434932279230449671459498768\ 2088521836539083349389779610268118679016534387026461349150269018064493460106902\ 0808141049756862424807807736557725580724225848787342173847251767343411512929985\ 469450688809606753317867696371928981622572883480529282593026 --------------------------------------------------- [A,B,C]=, [2, 2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | f(x, 1) - y f(x, y)| f(x, y) = x (2 + f(x, y)) |2 + --------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 4 x (-1 + 18 x) + (1 - 24 x + 96 x ) P + 32 x P = 0 and in Maple input notation 4*x*(-1+18*x)+(1-24*x+96*x^2)*P+32*x^2*P^2 = 0 The first, 30, terms, for the sake of the OEIS are [4, 24, 192, 1792, 18432, 202752, 2342912, 28114944, 347602944, 4402970624, 56899928064, 747827625984, 9971035013120, 134608972677120, 1837016568299520, 25310006052126720, 351675873566392320, 4923462229929492480, 69397372383768084480, 984180917442529198080, 14035275692223894650880, 201172284921875823329280, 2896880902875011855941632, 41893354595423248378232832 , 608229444496515309787676672, 8862771905520651656906145792, 129579837515198493190627786752, 1900504283556244566795874205696, 27955804945214436208352214122496, 412348122941912934073195158306816] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 8 (1 + 2 n) A[n] - ---------------- + A[n + 1] = 0 3 + n and in Maple input format -8*(1+2*n)/(3+n)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 2333475483627032495549431510743499867551258698568780011845528572996772435117368\ 2983669670014333044860293878594160607787173492395151529706415667027837887904184\ 0572335509842541606997312077940089884860227137870093130130563814078832006536682\ 7497160916603261582201399627401463002858870380993271772155818170580206506562196\ 4216862949686002617780073571602808806917243243633645859670050272624273134531313\ 9468357545523255001749861738788454805643393266647254484595844171882508026747289\ 60 --------------------------------------------------- [A,B,C]=, [2, 2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | 2 (f(x, 1) - y f(x, y))| f(x, y) = x (2 + f(x, y)) |2 + ------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 4 x (1 - 36 x + 196 x ) + (-1 + 46 x - 540 x + 2184 x ) P 2 2 3 3 + 4 x (1 - 72 x + 480 x ) P + 512 x P = 0 and in Maple input notation 4*x*(1-36*x+196*x^2)+(-1+46*x-540*x^2+2184*x^3)*P+4*x*(1-72*x+480*x^2)*P^2+512* x^3*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [4, 40, 528, 8096, 136512, 2458752, 46488832, 912380416, 18443166720, 381871974400, 8065581355008, 173231332958208, 3774218196172800, 83251463040565248, 1856254467836805120, 41783361106404835328, 948470953187075162112, 21692460276989211181056, 499489700776150324740096, 11571617328963955772620800, 269567408677774581640986624, 6311531310565800780855508992, 148460587643505824128196149248, 3506987228493457204067293986816, 83168760080208722410164560330752, 1979529059751087353682616146460672, 47274344906682627969753851627569152, 1132529129033409767489226730343759872, 27210841892282814431630664762478559232, 655572547629818268995178756532900724736] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 864 n (3 n + 7) (1 + 2 n) A[n] 12 (3 n + 5) (27 n + 99 n + 44) A[n + 1] - ------------------------------ + ----------------------------------------- (n + 5) (n + 4) (3 n + 4) (n + 5) (n + 4) (3 n + 4) 2 6 (18 n + 69 n + 55) A[n + 2] - ------------------------------ + A[3 + n] = 0 (n + 5) (3 n + 4) and in Maple input format -864*n*(3*n+7)*(1+2*n)/(n+5)/(n+4)/(3*n+4)*A[n]+12*(3*n+5)*(27*n^2+99*n+44)/(n+ 5)/(n+4)/(3*n+4)*A[n+1]-6*(18*n^2+69*n+55)/(n+5)/(3*n+4)*A[n+2]+A[3+n] = 0 Just for fun, A[400], equals 2504142078205754085429060656640125363272645683836714538797881139630663765260437\ 8295282707996216984100660143023869912766771282257258639496860673565593153388984\ 1085468741480684573990101254729552640897986630059150891560896233734889032511942\ 1245342227620614082092362605988477837384188231508215972337386361347074036638738\ 5317751058966404763393455554191566611675142051879925794188602128450788658422333\ 3028686206251911873375935585971803718112488793676494145523684934555586379093493\ 3562593038221852379861533479869468084646140155743357382731276893151970174656717\ 45798144 --------------------------------------------------- --------------------------------------------------- Let's take a=1, b=0 --------------------------------------------------- [A,B,C]=, [1, 1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / y f(x, 1) - y f(x, y)\ f(x, y) = x (1 + f(x, y)) |y + ---------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 x (-1 + 8 x) + (1 - 10 x + 12 x ) P + 2 x (1 + 3 x) P + x P = 0 and in Maple input notation x*(-1+8*x)+(1-10*x+12*x^2)*P+2*x*(1+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,C]=, [1, 2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 (y f(x, 1) - y f(x, y))\ f(x, y) = x (1 + f(x, y)) |2 y + -------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 2 x (-1 + 16 x) + (1 - 20 x + 48 x ) P + 4 x (1 + 6 x) P + 4 x P = 0 and in Maple input notation 2*x*(-1+16*x)+(1-20*x+48*x^2)*P+4*x*(1+6*x)*P^2+4*x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [2, 8, 48, 352, 2912, 26112, 248064, 2461184, 25259520, 266373120, 2872197120, 31549980672, 352057720832, 3981894221824, 45566999199744, 526822739410944, 6146265293127680, 72286374741934080, 856315516173680640, 10210145283436707840, 122456985608745123840, 1476582782992404971520, 17891784998173928325120, 217768583120631241900032, 2661506529451696761864192, 32652277170547860483080192 , 402004659190641191402078208, 4965596570692384661855797248, 61522899884341240471467589632, 764432604803004646281122414592] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 3 (3 n + 2) (3 n + 1) A[n] - -------------------------- + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -3*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 2068143393326921405269750356872501295051129703394418303954560324323840598666388\ 3348336784021615417956140092751003506710107388422118802932876088601508706344976\ 9535311983125245709535338629996639735298811423939470697271399405965879107094558\ 2085539836895028423730510715936597321655351186578091617841077946668633831132601\ 4332278913205231914104715248682166975076439873825792900056409380663917584537238\ 024953454330560977586323861443091471709504477855744 --------------------------------------------------- [A,B,C]=, [2, 2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / y f(x, 1) - y f(x, y)\ f(x, y) = x (2 + f(x, y)) |2 y + ---------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 4 x (-1 + 16 x) + (1 - 20 x + 48 x ) P + 2 x (1 + 6 x) P + x P = 0 and in Maple input notation 4*x*(-1+16*x)+(1-20*x+48*x^2)*P+2*x*(1+6*x)*P^2+x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [4, 16, 96, 704, 5824, 52224, 496128, 4922368, 50519040, 532746240, 5744394240, 63099961344, 704115441664, 7963788443648, 91133998399488, 1053645478821888, 12292530586255360, 144572749483868160, 1712631032347361280, 20420290566873415680, 244913971217490247680, 2953165565984809943040, 35783569996347856650240, 435537166241262483800064, 5323013058903393523728384, 65304554341095720966160384, 804009318381282382804156416, 9931193141384769323711594496, 123045799768682480942935179264, 1528865209606009292562244829184] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 3 (3 n + 2) (3 n + 1) A[n] - -------------------------- + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -3*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 4136286786653842810539500713745002590102259406788836607909120648647681197332776\ 6696673568043230835912280185502007013420214776844237605865752177203017412689953\ 9070623966250491419070677259993279470597622847878941394542798811931758214189116\ 4171079673790056847461021431873194643310702373156183235682155893337267662265202\ 8664557826410463828209430497364333950152879747651585800112818761327835169074476\ 049906908661121955172647722886182943419008955711488 --------------------------------------------------- --------------------------------------------------- Let's take a=1, b=1 --------------------------------------------------- [A,B,C]=, [1, 1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | y f(x, 1) - y f(x, y)| f(x, y) = x (1 + f(x, y)) |y + ----------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 2 2 x (-1 + 11 x + x ) + (1 - 14 x + 25 x + 4 x ) P + x (3 + 17 x + 6 x ) P 2 3 3 4 + x (3 + 4 x) P + x P = 0 and in Maple input notation x*(-1+11*x+x^2)+(1-14*x+25*x^2+4*x^3)*P+x*(3+17*x+6*x^2)*P^2+x^2*(3+4*x)*P^3+x^ 3*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855, 514956972502029690, 4384387181372914755, 37495248874510995828, 321973371914323166604, 2775265555241813680074, 24005427634602793982722, 208318056954199178840392, 1813260799116658123072932, 15827868548437082733879976 ] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: (4 n + 5) (1 + 2 n) (4 n + 3) A[n] -8/3 ---------------------------------- + A[n + 1] = 0 (3 n + 5) (3 n + 4) (n + 2) and in Maple input format -8/3*(4*n+5)*(1+2*n)*(4*n+3)/(3*n+5)/(3*n+4)/(n+2)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 7170696835284260895120413460923264346941661295908127975452333342840814621930471\ 0933702483917252670340673190154700578398032915966870654252986590816731490495894\ 5285950108557858391511285234137260805399659753596460594735608300602429368942305\ 4727771017630436437924032279057390489767579997987059886279244034544329091396452\ 04586900060111129184709460599147224924378846656845162920647873804080 --------------------------------------------------- [A,B,C]=, [1, 2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | 2 (y f(x, 1) - y f(x, y))| f(x, y) = x (1 + f(x, y)) |2 y + --------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 2 x (-1 + 22 x + 4 x ) + (1 - 28 x + 100 x + 32 x ) P 2 2 2 3 3 4 + 2 x (3 + 34 x + 24 x ) P + 4 x (3 + 8 x) P + 8 x P = 0 and in Maple input notation 2*x*(-1+22*x+4*x^2)+(1-28*x+100*x^2+32*x^3)*P+2*x*(3+34*x+24*x^2)*P^2+4*x^2*(3+ 8*x)*P^3+8*x^3*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [2, 12, 104, 1088, 12768, 161920, 2171520, 30379008, 439273472, 6522760192, 98992478208, 1530009141248, 24016345579520, 382027450613760, 6147484897935360, 99930208290471936, 1639010121656303616, 27097255057558601728, 451198595318436331520, 7561402643852789022720, 127457664437508070440960, 2159886089593153136885760, 36778905384762283697111040, 629065889341427871381454848, 10803633613709866519838588928, 186244918710607361369425575936, 3221953956784801190813018095616, 55919952611534427686846177738752, 973486978915609040926373245353984, 16995044445231065563915192027316224] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: (4 n + 5) (1 + 2 n) (4 n + 3) A[n] -16/3 ---------------------------------- + A[n + 1] = 0 (3 n + 5) (3 n + 4) (n + 2) and in Maple input format -16/3*(4*n+5)*(1+2*n)*(4*n+3)/(3*n+5)/(3*n+4)/(n+2)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 1851653102871096394054957651396571927492095776694386462127717326316466277735666\ 4240032856619292337799282649401344809270105204742247575354539523910748995838930\ 7589855444482424957114151290592984789250942057348094018219128862691905997568328\ 6357222310617203849135340222031887398062541241010998650439612239125150429253707\ 3955983506391837698200845625153070014171850999953046371401767641051882265117282\ 1683724063562363327711861377919245691647607219593698183648946654863643887453484\ 2729579296863174542793721774080 --------------------------------------------------- [A,B,C]=, [2, 2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 \ | y f(x, 1) - y f(x, y)| f(x, y) = x (2 + f(x, y)) |2 y + ----------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 4 x (-1 + 22 x + 4 x ) + (1 - 28 x + 100 x + 32 x ) P 2 2 2 3 3 4 + x (3 + 34 x + 24 x ) P + x (3 + 8 x) P + x P = 0 and in Maple input notation 4*x*(-1+22*x+4*x^2)+(1-28*x+100*x^2+32*x^3)*P+x*(3+34*x+24*x^2)*P^2+x^2*(3+8*x) *P^3+x^3*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [4, 24, 208, 2176, 25536, 323840, 4343040, 60758016, 878546944, 13045520384, 197984956416, 3060018282496, 48032691159040, 764054901227520, 12294969795870720 , 199860416580943872, 3278020243312607232, 54194510115117203456, 902397190636872663040, 15122805287705578045440, 254915328875016140881920, 4319772179186306273771520, 73557810769524567394222080, 1258131778682855742762909696, 21607267227419733039677177856, 372489837421214722738851151872, 6443907913569602381626036191232, 111839905223068855373692355477504, 1946973957831218081852746490707968, 33990088890462131127830384054632448] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: (4 n + 5) (1 + 2 n) (4 n + 3) A[n] -16/3 ---------------------------------- + A[n + 1] = 0 (3 n + 5) (3 n + 4) (n + 2) and in Maple input format -16/3*(4*n+5)*(1+2*n)*(4*n+3)/(3*n+5)/(3*n+4)/(n+2)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 3703306205742192788109915302793143854984191553388772924255434652632932555471332\ 8480065713238584675598565298802689618540210409484495150709079047821497991677861\ 5179710888964849914228302581185969578501884114696188036438257725383811995136657\ 2714444621234407698270680444063774796125082482021997300879224478250300858507414\ 7911967012783675396401691250306140028343701999906092742803535282103764530234564\ 3367448127124726655423722755838491383295214439187396367297893309727287774906968\ 5459158593726349085587443548160 Let's take a=2, b=2 --------------------------------------------------- [A,B,C]=, [1, 1, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 3 \ | 2 y f(x, 1) - y f(x, y)| f(x, y) = x (1 + f(x, y)) |y + -----------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 x (-1 + 16 x) + (1 - 20 x + 48 x ) P + 8 x (1 + 6 x) P + 16 x P = 0 and in Maple input notation x*(-1+16*x)+(1-20*x+48*x^2)*P+8*x*(1+6*x)*P^2+16*x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320, 5105072641718353920, 61228492804372561920, 738291391496202485760, 8945892499086964162560, 108884291560315620950016, 1330753264725848380932096, 16326138585273930241540096 , 201002329595320595701039104, 2482798285346192330927898624, 30761449942170620235733794816, 382216302401502323140561207296] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 3 (3 n + 2) (3 n + 1) A[n] - -------------------------- + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -3*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 1034071696663460702634875178436250647525564851697209151977280162161920299333194\ 1674168392010807708978070046375501753355053694211059401466438044300754353172488\ 4767655991562622854767669314998319867649405711969735348635699702982939553547279\ 1042769918447514211865255357968298660827675593289045808920538973334316915566300\ 7166139456602615957052357624341083487538219936912896450028204690331958792268619\ 012476727165280488793161930721545735854752238927872 --------------------------------------------------- [A,B,C]=, [1, 2, 2] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 3 \ | 2 2 (y f(x, 1) - y f(x, y))| f(x, y) = x (1 + f(x, y)) |2 y + ---------------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 2 x (-1 + 32 x) + (1 - 40 x + 192 x ) P + 16 x (1 + 12 x) P + 64 x P = 0 and in Maple input notation 2*x*(-1+32*x)+(1-40*x+192*x^2)*P+16*x*(1+12*x)*P^2+64*x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440 , 2941129850880, 64614360416256, 1442028424527872, 32619677465182208, 746569714888605696, 17262927525017812992, 402801642250415636480, 9474719710174783733760, 224477974671833337692160, 5353056650362464680017920, 128405456141675526975651840, 3096618536518088070833111040, 75043585384980900269764116480, 1826775278514392200852543635456, 44652669930021478140296111849472, 1095628613964300587325001453010944, 26978076000991089786600035778035712, 666471089882923256216333370255212544, 16514927684895488145564057412086792192, 410401629703124684789183599105649147904 ] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 6 (3 n + 2) (3 n + 1) A[n] - -------------------------- + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -6*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 2670231512642344119196825536652428850024390148397221733124566614098682733113425\ 1642669913258035137536716846028016248932782109253521666422813580816966795060345\ 5970402847079931524999474053729944474419617244105939609950358143354495345146697\ 5779951216458866295251743754713831792451779705913364942766268450431032060703092\ 5220609949976069630669285013359048582302890426756787098803916929424647135535326\ 9618559954330217093444725701512249812493384995758010893912319610507275629758373\ 8763063573445662604658075269292372934404905105222360996195373090548249269431672\ 3337661775872 --------------------------------------------------- [A,B,C]=, [2, 2, 1] Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation / 2 3 \ | 2 y f(x, 1) - y f(x, y)| f(x, y) = x (2 + f(x, y)) |2 y + -----------------------| \ 1 - y / Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 4 x (-1 + 32 x) + (1 - 40 x + 192 x ) P + 8 x (1 + 12 x) P + 16 x P = 0 and in Maple input notation 4*x*(-1+32*x)+(1-40*x+192*x^2)*P+8*x*(1+12*x)*P^2+16*x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [4, 32, 384, 5632, 93184, 1671168, 31752192, 630063104, 12932874240, 272766074880, 5882259701760, 129228720832512, 2884056849055744, 65239354930364416, 1493139429777211392, 34525855050035625984, 805603284500831272960, 18949439420349567467520, 448955949343666675384320, 10706113300724929360035840, 256810912283351053951303680, 6193237073036176141666222080, 150087170769961800539528232960, 3653550557028784401705087270912, 89305339860042956280592223698944, 2191257227928601174650002906021888, 53956152001982179573200071556071424, 1332942179765846512432666740510425088, 33029855369790976291128114824173584384, 820803259406249369578367198211298295808] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 6 (3 n + 2) (3 n + 1) A[n] - -------------------------- + A[n + 1] = 0 (n + 2) (2 n + 3) and in Maple input format -6*(3*n+2)*(3*n+1)/(n+2)/(2*n+3)*A[n]+A[n+1] = 0 Just for fun, A[400], equals 5340463025284688238393651073304857700048780296794443466249133228197365466226850\ 3285339826516070275073433692056032497865564218507043332845627161633933590120691\ 1940805694159863049998948107459888948839234488211879219900716286708990690293395\ 1559902432917732590503487509427663584903559411826729885532536900862064121406185\ 0441219899952139261338570026718097164605780853513574197607833858849294271070653\ 9237119908660434186889451403024499624986769991516021787824639221014551259516747\ 7526127146891325209316150538584745868809810210444721992390746181096498538863344\ 6675323551744 This ends this file, that took, 85.500, seconds.