--------------------------------------- For a list of the SCHUTZENBERGER procedures type ezraS();, for help with a specific procedure, type ezra(procedure_name); . --------------------------------------- For a list of the MAIN procedures type ezra();, for help with a specific procedure, type ezra(procedure_name); . --------------------------------------- Enumeration of Alpha(a,b) Description Trees for a and b up to, 6 By Shalosh B. Ekhad In this article, we will try to enumerate Alpha(a,b) description trees for a\ ,b up to , 6 by discovering (easily provable, once discovered!) algebraic equations for t\ he generating functions of the enumerating sequence, and will also present, in each case, the implie\ d linear recurrence equation with polynomial coefficients satisfied by the enumerating sequence in each case, provided its order + deg\ ree are <=, 10 We will do this by using as TRAINING DATA (so to speak!) the first , 40, terms, and admit failure otherwise For the sake of the OEIS, we will give the first, 30, terms in each case, and just for fun the, 100, -th term. Finally we will output the set of all successful pairs [a,b]. According to the article Cori, Robert, Benjamin Jacquard, and Gilles Schaeffer. "Description trees fo\ r some families of planar maps" Proceedings of the 9th Conference on Formal Power Series and Algebraic Combi\ natorics. 1997. Available from http://www-igm.univ-mlv.fr/~fpsac/FPSAC97/ARTICLES/Schaeffer\ .ps.gz [viewed Dec. 24, 2014] The generating function, f(x,y),in the two variables (y being the so-called \ catalytic variable taking care of the label of the root) satisfies the FUNCTIONAL EQUATION a (b + 1) b x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------------- (1 - f(x, y)) (1 - y) we would be interested in the straight enumeration, whose generating functio\ n is f(x,1) -------------------------------------------------------- For the Enumeration of , Alpha(0, 0), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation x (f(x, 1) - y f(x, y)) f(x, y) = x + ----------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 -P + P + x = 0 and in Maple input notation -P+P^2+x = 0 The first, 30, terms, for the sake of the OEIS are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 (-1 + 2 n) A[n] - ----------------- + A[1 + n] = 0 1 + n and in Maple input format -2*(-1+2*n)/(1+n)*A[n]+A[1+n] = 0 Just for fun, A[100], equals 227508830794229349661819540395688853956041682601541047340 -------------------------------------------------------- For the Enumeration of , Alpha(0, 1), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 2 x (f(x, 1) - y f(x, y)) f(x, y) = y x + ------------------------ (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 4 5 -x (-1 + 11 x + x ) + (-1 + 13 x - 20 x + 62 x + 17 x + x ) P 2 3 4 2 2 3 + (1 - 12 x - 58 x + 84 x + x ) P + 64 x P = 0 and in Maple input notation -x*(-1+11*x+x^2)+(-1+13*x-20*x^2+62*x^3+17*x^4+x^5)*P+(1-12*x-58*x^2+84*x^3+x^4 )*P^2+64*x^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 12, 61, 354, 2235, 14964, 104637, 756642, 5619587, 42657804, 329751837, 2588548194, 20589888699, 165661020468, 1346288408957, 11038223459010, 91217892269379, 759143460388812, 6358097461220733, 53558660584500834, 453530448715334779, 3858882926111838516, 32977952584840092477, 282970834845053622882, 2437147645223408797251, 21063241765653674080332, 182627208670377440748125, 1588207532170508359312482, 13850472766522491009040635 ] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: (n - 1) (n - 2) (3 n + 8) A[n] ------------------------------ (3 n + 2) (n + 8) (n + 6) 2 3 (n - 1) (3 n + 8) (18 n + 39 n + 10) A[1 + n] - ------------------------------------------------ (3 n + 5) (3 n + 2) (n + 8) (n + 6) 3 2 (3 n + 4) (963 n + 4815 n + 7462 n + 3880) A[n + 2] - ----------------------------------------------------- (3 n + 5) (3 n + 2) (n + 8) (n + 6) 2 3 4 6 (4480 + 12486 n + 10257 n + 3339 n + 378 n ) A[n + 3] - --------------------------------------------------------- (3 n + 5) (3 n + 2) (n + 8) (n + 6) 2 3 4 (23280 + 58558 n + 42201 n + 11322 n + 999 n ) A[n + 4] + --------------------------------------------------------- (3 n + 5) (3 n + 2) (n + 8) (n + 6) 3 2 3 (18 n + 237 n + 929 n + 1000) A[n + 5] - ------------------------------------------ + A[n + 6] = 0 (3 n + 5) (n + 8) (n + 6) and in Maple input format (n-1)*(n-2)*(3*n+8)/(3*n+2)/(n+8)/(n+6)*A[n]-3*(n-1)*(3*n+8)*(18*n^2+39*n+10)/( 3*n+5)/(3*n+2)/(n+8)/(n+6)*A[1+n]-(3*n+4)*(963*n^3+4815*n^2+7462*n+3880)/(3*n+5 )/(3*n+2)/(n+8)/(n+6)*A[n+2]-6*(4480+12486*n+10257*n^2+3339*n^3+378*n^4)/(3*n+5 )/(3*n+2)/(n+8)/(n+6)*A[n+3]+(23280+58558*n+42201*n^2+11322*n^3+999*n^4)/(3*n+5 )/(3*n+2)/(n+8)/(n+6)*A[n+4]-3*(18*n^3+237*n^2+929*n+1000)/(3*n+5)/(n+8)/(n+6)* A[n+5]+A[n+6] = 0 Just for fun, A[100], equals 1566741013357158844895557009202820852277760585341785945711250223261086127863842\ 04585655752125 -------------------------------------------------------- For the Enumeration of , Alpha(0, 2), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 3 2 x (f(x, 1) - y f(x, y)) f(x, y) = x y + ------------------------ (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 3 x (-1 + 16 x) + (1 - 20 x + 62 x - 184 x ) P 2 3 4 5 2 + (-1 + 18 x + 159 x - 575 x + 616 x - 16 x ) P 2 2 3 4 3 3 2 4 + x (-180 - 307 x + 1326 x - 552 x + 64 x ) P + 27 x (x - 4) P = 0 and in Maple input notation x*(-1+16*x)+(1-20*x+62*x^2-184*x^3)*P+(-1+18*x+159*x^2-575*x^3+616*x^4-16*x^5)* P^2+x^2*(-180-307*x+1326*x^2-552*x^3+64*x^4)*P^3+27*x^3*(x-4)^2*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [1, 5, 30, 220, 1835, 16656, 160450, 1615475, 16830720, 180193510, 1972602914, 21998122770, 249193305445, 2861010554885, 33232125358680, 389961573609661, 4617336942389990, 55110581657284815, 662503220558892910, 8015692353360579230, 97550335451689838706, 1193498626656068716180, 14673073411732727622680, 181197293855390193793245, 2246777663356787943570010, 27964706039748023255389658 , 349286053885925266429057170, 4376883896212114208991797470, 55012920530956635062480611055, 693414667715083155399524150640] -------------------------------------------------------- For the Enumeration of , Alpha(1, 0), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation x (y f(x, 1) - y f(x, y)) f(x, y) = x + ------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation P - x = 0 and in Maple input notation P-x = 0 The first, 30, terms, for the sake of the OEIS are [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: A[n] = 0 and in Maple input format A[n] = 0 Just for fun, A[100], equals 0 -------------------------------------------------------- For the Enumeration of , Alpha(1, 1), trees we have the following 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, y) = y x + -------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 -x (-1 + 11 x + x ) + (x + 1) (x + 12 x - 1) P 2 3 4 2 2 3 2 4 + (-20 x - 10 x + 20 x + x + 1) P + 8 x (x + 1) P + 16 x P = 0 and in Maple input notation -x*(-1+11*x+x^2)+(x^2+1)*(x^2+12*x-1)*P+(-20*x-10*x^2+20*x^3+x^4+1)*P^2+8*x*(x^ 2+1)*P^3+16*x^2*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [1, 2, 7, 32, 169, 978, 6031, 38976, 261137, 1800434, 12705975, 91419808, 668590169, 4958383202, 37218815391, 282335155456, 2161739386785, 16688811044514 , 129792922660967, 1016149466420640, 8003298699439625, 63379255346833842, 504411545081156911, 4032754800688091712, 32377140760432065969, 260948294945780312338, 2110691896559696621079, 17129243799824815641376, 139442334858660220397753, 1138424871757179998327746] -------------------------------------------------------- For the Enumeration of , Alpha(1, 2), trees we have the following 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, y) = x y + -------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 3 2 x (-1 + 16 x) + (-32 x - 19 x + 32 x + 1) P 3 4 2 2 2 3 3 + (-98 x + 16 x + 30 x + 2 x - 1) P - x (91 x + 24 x + 12 - 13 x) P 2 2 3 4 3 5 + 3 x (24 x - 44 x + 9 x - 16) P + 16 x (x - 4) P = 0 and in Maple input notation x*(-1+16*x)+(-32*x^3-19*x+32*x^2+1)*P+(-98*x^3+16*x^4+30*x+2*x^2-1)*P^2-x*(91*x ^2+24*x^3+12-13*x)*P^3+3*x^2*(24*x-44*x^2+9*x^3-16)*P^4+16*x^3*(x-4)*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [1, 4, 22, 152, 1205, 10444, 96354, 931008, 9322266, 96029256, 1012320228, 10878432504, 118809165117, 1315687502364, 14745676482010, 167004987303872, 1909007458697534, 22001414831756344, 255435437201431444, 2985238112064077680, 35096990171001004034, 414874953348983882488, 4928507292861617909332, 58814532699342181628160, 704801528945525754611460, 8478561146979712211763024, 102359083462329816874773696, 1239848625502365321079074680, 15064323036607006655532351821, 183560761757757835281797176764] -------------------------------------------------------- For the Enumeration of , Alpha(2, 0), trees we have the following 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, y) = x + -------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 x (4 x - 1) + (-1 + 6 x + 15 x - 60 x ) P 2 3 4 2 2 3 + (1 - 24 x + 18 x + 48 x - 27 x ) P + 16 x (x - 1) P = 0 and in Maple input notation x*(4*x-1)^2+(-1+6*x+15*x^2-60*x^3)*P+(1-24*x+18*x^2+48*x^3-27*x^4)*P^2+16*x*(x-\ 1)^2*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, -1, -1, 0, 3, 8, 9, -17, -117, -303, -249, 1467, 7863, 18763, 5723, -149796 , -679401, -1416256, 665133, 17019032, 66682711, 114484674, -194096915, -\ 2056109254, -7003776741, -8974181072, 35799305181, 257460926022, 759990525047, 570806227048] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 2 54 (n - 1) (2 n + 1) (12 n + 87 n + 130) A[n] - ---------------------------------------------- (n + 7) (n + 6) %1 2 3 4 3 (-520 + 4071 n + 13758 n + 7479 n + 972 n ) A[1 + n] + -------------------------------------------------------- (n + 7) (n + 6) %1 2 3 4 9 (11813 n + 10006 n + 3177 n + 324 n + 4360) A[n + 2] - --------------------------------------------------------- (n + 7) (n + 6) %1 4 2 3 6 (252 n + 17080 n + 10968 n + 2835 n + 8625) A[n + 3] + --------------------------------------------------------- (n + 7) (n + 6) %1 2 3 4 6 (4863 n + 2329 n + 477 n + 36 n + 3495) A[n + 4] - ----------------------------------------------------- (n + 7) (n + 6) %1 2 (n + 5) (12 n + 99 n + 49) A[n + 5] - ------------------------------------ + A[n + 6] = 0 (n + 7) %1 2 %1 := 12 n + 63 n + 55 and in Maple input format -54*(n-1)*(2*n+1)*(12*n^2+87*n+130)/(n+7)/(n+6)/(12*n^2+63*n+55)*A[n]+3*(-520+ 4071*n+13758*n^2+7479*n^3+972*n^4)/(n+7)/(n+6)/(12*n^2+63*n+55)*A[1+n]-9*(11813 *n+10006*n^2+3177*n^3+324*n^4+4360)/(n+7)/(n+6)/(12*n^2+63*n+55)*A[n+2]+6*(252* n^4+17080*n+10968*n^2+2835*n^3+8625)/(n+7)/(n+6)/(12*n^2+63*n+55)*A[n+3]-6*( 4863*n+2329*n^2+477*n^3+36*n^4+3495)/(n+7)/(n+6)/(12*n^2+63*n+55)*A[n+4]-(n+5)* (12*n^2+99*n+49)/(n+7)/(12*n^2+63*n+55)*A[n+5]+A[n+6] = 0 Just for fun, A[100], equals -217142047442471947653232199746693861493531727136218 -------------------------------------------------------- For the Enumeration of , Alpha(2, 1), trees we have the following 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, y) = y x + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 x (-1 + 11 x + x ) + (1 - 12 x - 8 x ) P + 16 x P = 0 and in Maple input notation x*(-1+11*x+x^2)+(1-12*x-8*x^2)*P+16*x*P^2 = 0 The first, 30, terms, for the sake of the OEIS are [1, 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] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 4 (-1 + 2 n) A[n] - ----------------- + A[1 + n] = 0 n + 2 and in Maple input format -4*(-1+2*n)/(n+2)*A[n]+A[1+n] = 0 Just for fun, A[100], equals 2855462434787401147760366551595005675901125247669571408592901790698719724267466\ 915840 -------------------------------------------------------- For the Enumeration of , Alpha(2, 2), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 2 3 2 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 3 2 -x (-1 + 16 x) + (-1 + 18 x - 8 x ) P + (1 - 37 x + 16 x + 16 x ) P 2 3 2 4 2 5 + 2 x (9 - 8 x + 16 x ) P + x (4 x + 1) P + 16 x P = 0 and in Maple input notation -x*(-1+16*x)+(-1+18*x-8*x^2)*P+(1-37*x+16*x^2+16*x^3)*P^2+2*x*(9-8*x+16*x^2)*P^ 3+x*(4*x+1)^2*P^4+16*x^2*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 15, 97, 726, 5967, 52345, 481791, 4601256, 45249018, 455696598, 4680680199, 48883283148, 517818481221, 5552941699287, 60189363294617, 658584922147188, 7266707288536428, 80781110248307576, 904064886605846346, 10179488103710373624, 115251488520344315046, 1311448041047286565746, 14991870510202402392795, 172107289107551774355466, 1983529865045931349851732, 22942861760283146188431144, 266264221640530811291723841, 3099796757355812472638041872, 36192435300402828139718408067] -------------------------------------------------------- For the Enumeration of , Alpha(3, 0), trees we have the following 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, y) = x + -------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 3 2 3 4 5 -x (4 x - 1) + (9 x + 12 x - 165 x + 224 x + 128 x - 1) P 2 3 4 6 5 2 + (-39 x + 81 x + 95 x - 378 x - 64 x + 1 + 240 x ) P 4 3 + 27 x (x - 1) P = 0 and in Maple input notation -x*(4*x-1)^3+(9*x+12*x^2-165*x^3+224*x^4+128*x^5-1)*P+(-39*x+81*x^2+95*x^3-378* x^4-64*x^6+1+240*x^5)*P^2+27*x*(x-1)^4*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, -2, -1, 4, 13, 10, -69, -336, -567, 1590, 13815, 38876, -20731, -662302, -\ 2736717, -2454392, 32111809, 192975886, 393045231, -1360077132, -13369076483, -\ 41869888070, 28961679787, 888351580256, 3898582132153, 3364519283686, -\ 54419770283545, -335842984836212, -685518753153803, 2803034349779090] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 512 (n - 1) (315527 n + 924140) (2 n + 3) A[n] ---------------------------------------------- (n + 8) (n + 7) (151927 n + 294401) 2 3 256 (-1386210 - 391876 n + 6542731 n + 2197016 n ) A[1 + n] - ------------------------------------------------------------ (n + 8) (n + 7) (151927 n + 294401) 2 3 8 (606125379 n + 306276964 + 358053091 n + 59389826 n ) A[n + 2] + ----------------------------------------------------------------- (n + 8) (n + 7) (151927 n + 294401) 2 3 2 (1594431748 + 2475716025 n + 1126215062 n + 156421461 n ) A[n + 3] - --------------------------------------------------------------------- (n + 8) (n + 7) (151927 n + 294401) 2 3 (840039468 n + 2284602852 + 2519187353 n + 87987655 n ) A[n + 4] + ----------------------------------------------------------------- (n + 8) (n + 7) (151927 n + 294401) 2 3 (130573282 n + 11207401 n + 527474472 + 478377405 n) A[n + 5] - --------------------------------------------------------------- (n + 8) (n + 7) (151927 n + 294401) 2 (128581 n + 1072957 n + 3043148) A[n + 6] + ------------------------------------------ + A[n + 7] = 0 (n + 8) (151927 n + 294401) and in Maple input format 512*(n-1)*(315527*n+924140)*(2*n+3)/(n+8)/(n+7)/(151927*n+294401)*A[n]-256*(-\ 1386210-391876*n+6542731*n^2+2197016*n^3)/(n+8)/(n+7)/(151927*n+294401)*A[1+n]+ 8*(606125379*n+306276964+358053091*n^2+59389826*n^3)/(n+8)/(n+7)/(151927*n+ 294401)*A[n+2]-2*(1594431748+2475716025*n+1126215062*n^2+156421461*n^3)/(n+8)/( n+7)/(151927*n+294401)*A[n+3]+(840039468*n^2+2284602852+2519187353*n+87987655*n ^3)/(n+8)/(n+7)/(151927*n+294401)*A[n+4]-(130573282*n^2+11207401*n^3+527474472+ 478377405*n)/(n+8)/(n+7)/(151927*n+294401)*A[n+5]+(128581*n^2+1072957*n+3043148 )/(n+8)/(151927*n+294401)*A[n+6]+A[n+7] = 0 Just for fun, A[100], equals -33575917104609983877840438976210232785256528980156212260608732 -------------------------------------------------------- For the Enumeration of , Alpha(3, 1), trees we have the following 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, y) = y x + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation P - x = 0 and in Maple input notation P-x = 0 The first, 30, terms, for the sake of the OEIS are [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: A[n] = 0 and in Maple input format A[n] = 0 Just for fun, A[100], equals 0 -------------------------------------------------------- For the Enumeration of , Alpha(3, 2), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 3 3 2 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 x (-1 + 16 x) + (1 - 18 x) P + 27 x P = 0 and in Maple input notation x*(-1+16*x)+(1-18*x)*P+27*x*P^2 = 0 The first, 30, terms, for the sake of the OEIS are [1, 2, 9, 54, 378, 2916, 24057, 208494, 1876446, 17399772, 165297834, 1602117468, 15792300756, 157923007560, 1598970451545, 16365932856990, 169114639522230, 1762352559231660, 18504701871932430, 195621134074714260, 2080697516976506220, 22254416920705240440, 239234981897581334730, 2583737804493878415084, 28023617725664373578988, 305146059679456512304536, 3334810509354060455899572, 36567922137054869826760824, 402247143507603568094369064, 4437694292890336138331426448] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 6 (-1 + 2 n) A[n] - ----------------- + A[1 + n] = 0 n + 2 and in Maple input format -6*(-1+2*n)/(n+2)*A[n]+A[1+n] = 0 Just for fun, A[100], equals 7739467799298257068789058467610299092251843633611894313571443312429302881766533\ 61311535025028208795560 -------------------------------------------------------- For the Enumeration of , Alpha(4, 0), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 4 x (y f(x, 1) - y f(x, y)) f(x, y) = x + -------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 4 27 x (4 x - 1) 5 6 2 3 4 + (-27 - 2792 x - 39600 x + 324 x - 2 x - 7764 x + 22485 x ) P + (27 4 5 6 7 8 2 - 1456 x - 35918 x + 64944 x - 42492 x + 18000 x - 3125 x + 4964 x 3 2 6 3 + 1968 x ) P + 1024 x (x - 1) P = 0 and in Maple input notation 27*x*(4*x-1)^4+(-27-2792*x^5-39600*x^6+324*x-2*x^2-7764*x^3+22485*x^4)*P+(27-\ 1456*x-35918*x^4+64944*x^5-42492*x^6+18000*x^7-3125*x^8+4964*x^2+1968*x^3)*P^2+ 1024*x*(x-1)^6*P^3 = 0 The first, 30, terms, for the sake of the OEIS are [1, -3, 0, 11, 21, -33, -322, -828, 1113, 18371, 66732, -24048, -1449457, -\ 6796131, -3443874, 133123907, 772980651, 970390059, -13187742316, -93851700483, -181557305343, 1347499440263, 11881204311486, 30640750438071, -136724320912849, -1545520705382097, -4969385541649128, 13102555037819807, 204484207092283185, 790974666322628643] -------------------------------------------------------- For the Enumeration of , Alpha(4, 2), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 4 3 2 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 3 x (-1 + 16 x) (x - 4) - 2 x (-8 + 126 x + 357 x + 92 x ) P 2 3 2 2 3 3 - x (1 - 649 x - 808 x + 16 x ) P + (438 x - 72 x + 192 x - 27) P 2 4 5 + (-1504 x - 768 x + 27) P + 1024 x P = 0 and in Maple input notation x^2*(-1+16*x)*(x-4)^2-2*x*(-8+126*x+357*x^2+92*x^3)*P-x*(1-649*x-808*x^2+16*x^3 )*P^2+(438*x-72*x^2+192*x^3-27)*P^3+(-1504*x-768*x^2+27)*P^4+1024*x*P^5 = 0 The first, 30, terms, for the sake of the OEIS are [0, 1, 1, 4, 22, 142, 1011, 7696, 61513, 510332, 4360403, 38156714, 340565453, 3090708068, 28450237535, 265121135692, 2497207594518, 23744355079734, 227666753984499, 2199300625049734, 21388809810288175, 209278487956406062, 2058999508833424824, 20359696208602230468, 202248836112227034779, 2017618358157114452448, 20206339119675350145831, 203097813771765619103534, 2048230873268156293808711, 20720844455756803257541796, 210232525346283748128556289] -------------------------------------------------------- For the Enumeration of , Alpha(4, 3), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 4 4 3 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 2 -x (27 - 587 x + 729 x ) + (27 - 614 x + 93 x ) P + (-54 + 2289 x - 1920 x ) P 2 3 4 + (27 - 2688 x + 1024 x ) P + 1024 x P = 0 and in Maple input notation -x*(27-587*x+729*x^2)+(27-614*x+93*x^2)*P+(-54+2289*x-1920*x^2)*P^2+(27-2688*x+ 1024*x^2)*P^3+1024*x*P^4 = 0 The first, 30, terms, for the sake of the OEIS are [1, 3, 18, 142, 1302, 13137, 141656, 1603998, 18856494, 228358117, 2832972018, 35854104588, 461461010280, 6025062720924, 79646874250512, 1064313912752390, 14358253138573086, 195342638293978557, 2677712354561624534, 36954719638839162966, 513134176442512374726, 7164770079514900720505, 100547549776719309849312, 1417598273726925766188552, 20071710542387226565948232 , 285312779511610472885499372, 4070389527468339899388534936, 58265938099941760120344045760, 836672623725675043973468192496, 12049436662499107932741299734632] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 640 (n - 1) (3 n - 1) (3 n + 1) (23063 n - 861504) A[n] 128 --- --------------------------------------------------- - --- 27 (n + 5) (n + 4) (390413 n + 494919) (2 n + 7) 81 2 3 4 (1615320 + 102060246 n + 235808342 n + 156860229 n + 32776603 n ) A[1 + n]/((n + 5) (n + 4) (390413 n + 494919) (2 n + 7)) + 5/54 4 2 3 (213936823 n + 2885337190 n + 2878433071 n + 1284109980 n + 1084858136) A[n + 2]/((n + 5) (n + 4) (390413 n + 494919) (2 n + 7)) 3 2 (239145297 n + 1679743983 n + 3722884051 n + 2537637405) A[n + 3] - 1/18 ------------------------------------------------------------------- (n + 5) (390413 n + 494919) (2 n + 7) + A[n + 4] = 0 and in Maple input format 640/27*(n-1)*(3*n-1)*(3*n+1)*(23063*n-861504)/(n+5)/(n+4)/(390413*n+494919)/(2* n+7)*A[n]-128/81*(1615320+102060246*n+235808342*n^2+156860229*n^3+32776603*n^4) /(n+5)/(n+4)/(390413*n+494919)/(2*n+7)*A[1+n]+5/54*(213936823*n^4+2885337190*n+ 2878433071*n^2+1284109980*n^3+1084858136)/(n+5)/(n+4)/(390413*n+494919)/(2*n+7) *A[n+2]-1/18*(239145297*n^3+1679743983*n^2+3722884051*n+2537637405)/(n+5)/( 390413*n+494919)/(2*n+7)*A[n+3]+A[n+4] = 0 Just for fun, A[100], equals 2653902816719570232673965301726453559970137239914907549418543573539765911658532\ 93763996486133917487790028268318936 -------------------------------------------------------- For the Enumeration of , Alpha(5, 2), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 5 3 2 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation P - x = 0 and in Maple input notation P-x = 0 The first, 30, terms, for the sake of the OEIS are [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: A[n] = 0 and in Maple input format A[n] = 0 Just for fun, A[100], equals 0 -------------------------------------------------------- For the Enumeration of , Alpha(5, 3), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 5 4 3 x (y f(x, 1) - y f(x, y)) f(x, y) = x y + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 2 -x (27 - 587 x + 729 x ) - x (918 x - 27 + 578 x) P 2 2 2 3 - x (486 x - 2768 x + 63) P + x (-4262 x + 729 x + 54) P 2 4 5 6 + (-64 + 675 x + 1418 x) P + (64 - 4625 x) P + 3125 x P = 0 and in Maple input notation -x^2*(27-587*x+729*x^2)-x*(918*x^2-27+578*x)*P-x*(486*x^2-2768*x+63)*P^2+x*(-\ 4262*x+729*x^2+54)*P^3+(-64+675*x^2+1418*x)*P^4+(64-4625*x)*P^5+3125*x*P^6 = 0 The first, 30, terms, for the sake of the OEIS are [0, 1, 2, 11, 82, 715, 6872, 70616, 762050, 8537559, 98526050, 1164679935, 14044415336, 172216554564, 2142174747664, 26977214593068, 343413634322162, 4413219343261887, 57193368492381350, 746788233645847097, 9817024968815967634, 129840444171888980435, 1726807530409338889568, 23081824345185455839040, 309958666900268952612088, 4180060178451847851878564, 56593033128598176087325784 , 768984862987467229343855988, 10484155946057113887671678144, 143386528635416136741843385032, 1966758450246135988756483031712] -------------------------------------------------------- For the Enumeration of , Alpha(5, 4), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 5 5 4 x (y f(x, 1) - y f(x, y)) f(x, y) = y x + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 x (1024 - 27833 x + 65536 x ) + (-1024 + 28857 x - 33068 x ) P 2 2 2 3 + (3072 - 138680 x + 237850 x ) P + (-3072 + 239528 x - 247500 x ) P 2 4 5 + (1024 - 180000 x + 84375 x ) P + 50000 x P = 0 and in Maple input notation x*(1024-27833*x+65536*x^2)+(-1024+28857*x-33068*x^2)*P+(3072-138680*x+237850*x^ 2)*P^2+(-3072+239528*x-247500*x^2)*P^3+(1024-180000*x+84375*x^2)*P^4+50000*x*P^ 5 = 0 The first, 30, terms, for the sake of the OEIS are [1, 4, 30, 292, 3289, 40692, 537540, 7452820, 107246532, 1589482888, 24128928534, 373634953212, 5883344986190, 93973865629368, 1519677511590144, 24841311513614772, 409936220451591276, 6821999445847764720, 114385307618565927960, 1930905466209914583152, 32794496739609265545081, 560075048685233363047252, 9613579047958144348248420, 165780218182715936815365324, 2870956366512828485425868124, 49914101633711023244080491232, 870954207666937678986281658804, 15248572177469636878049483365480, 267807723880599400451294972449932, 4717214340344336657696944813840944] Furthermore, the sequence of coefficients, let's call them , A[n], satisfy: 10125 ----- (n - 1) (4 n + 1) (2 n + 1) (4 n - 1) ( 1024 2 2292892006441497598378061035978512167 n + 11591906735047121406794521593602717311 n + 14326660593074199655071724051573320000) A[n]/((3 n + 11) (3 n + 10) (n + 5) (n + 4) %1) - 1/24576 (137535941693512316688688550895103872000000 + 89127394738029687714830700784807661988219288 n 2 + 311362437029863541121225480761535973909437838 n 3 + 414204022439215465500647653843486653657052873 n 4 + 264715975279282523826222511156818211061181717 n 5 + 80935192446973304484362550241862816886686015 n 6 + 9440631650061404023721686662380219453622269 n ) A[1 + n]/((3 n + 11) (3 n + 10) (n + 5) (n + 4) %1) + 1/3072 ( 11411155455146089850716948637494830141110640 + 40345338187082905907144154461755376625096012 n 2 + 57735447242165306713849085724764494862662915 n 3 + 42959683739055295731480665597513804683318873 n 4 + 17554596265589153280500035761950525382979355 n 5 + 3736023733253071137297104787462958179887699 n 6 + 323369361307569401544512217354056915440746 n ) A[n + 2]/((3 n + 11) (3 n + 10) (n + 5) (n + 4) %1) - 1/16 ( 5 206227290913649900647068172354616629440 n 4 + 2446808460036189623362912930496925078492 n 3 + 11427441718892126446277013600651501549098 n 2 + 26142101045444562824600747720599483135634 n + 29105361379920877308262699374966824210177 n + 12494047305826869923751973812169280304099) A[n + 3]/((n + 5) (3 n + 10) (3 n + 11) %1) + A[n + 4] = 0 2 %1 := 48359787393052753994800080242451894 n + 160440703570601759279887729596219323 n + 122908714342237696527264544054213803 and in Maple input format 10125/1024*(n-1)*(4*n+1)*(2*n+1)*(4*n-1)*(2292892006441497598378061035978512167 *n^2+11591906735047121406794521593602717311*n+ 14326660593074199655071724051573320000)/(3*n+11)/(3*n+10)/(n+5)/(n+4)/( 48359787393052753994800080242451894*n^2+160440703570601759279887729596219323*n+ 122908714342237696527264544054213803)*A[n]-1/24576*( 137535941693512316688688550895103872000000+ 89127394738029687714830700784807661988219288*n+ 311362437029863541121225480761535973909437838*n^2+ 414204022439215465500647653843486653657052873*n^3+ 264715975279282523826222511156818211061181717*n^4+ 80935192446973304484362550241862816886686015*n^5+ 9440631650061404023721686662380219453622269*n^6)/(3*n+11)/(3*n+10)/(n+5)/(n+4)/ (48359787393052753994800080242451894*n^2+160440703570601759279887729596219323*n +122908714342237696527264544054213803)*A[1+n]+1/3072*( 11411155455146089850716948637494830141110640+ 40345338187082905907144154461755376625096012*n+ 57735447242165306713849085724764494862662915*n^2+ 42959683739055295731480665597513804683318873*n^3+ 17554596265589153280500035761950525382979355*n^4+ 3736023733253071137297104787462958179887699*n^5+ 323369361307569401544512217354056915440746*n^6)/(3*n+11)/(3*n+10)/(n+5)/(n+4)/( 48359787393052753994800080242451894*n^2+160440703570601759279887729596219323*n+ 122908714342237696527264544054213803)*A[n+2]-1/16*( 206227290913649900647068172354616629440*n^5+ 2446808460036189623362912930496925078492*n^4+ 11427441718892126446277013600651501549098*n^3+ 26142101045444562824600747720599483135634*n^2+ 29105361379920877308262699374966824210177*n+ 12494047305826869923751973812169280304099)/(n+5)/(3*n+10)/(3*n+11)/( 48359787393052753994800080242451894*n^2+160440703570601759279887729596219323*n+ 122908714342237696527264544054213803)*A[n+3]+A[n+4] = 0 Just for fun, A[100], equals 1370916642635126114220128855149076169468373297902285389714303135936364785292921\ 07058190933983452065905960124919782898842192 -------------------------------------------------------- For the Enumeration of , Alpha(6, 5), trees we have the following Theorem: Let f(x,y) be (unique!) formal power series, in the variables, x,y,\ satisfying the FUNCTIONAL Equation 6 6 5 x (y f(x, 1) - y f(x, y)) f(x, y) = y x + --------------------------- (1 - f(x, y)) (1 - y) Then P(x)=f(x,1) satisfies the following algebraic equation 2 2 -x (84375 - 2752164 x + 9765625 x ) + (6402280 x - 2836539 x + 84375) P 2 2 + (-45201260 x + 16617915 x - 337500) P 2 3 + (69780960 x - 37663785 x + 506250) P 2 4 + (41957120 x - 47278080 x - 337500) P 2 5 6 + (-23094720 x + 11943936 x + 84375) P + 5038848 x P = 0 and in Maple input notation -x*(84375-2752164*x+9765625*x^2)+(6402280*x^2-2836539*x+84375)*P+(-45201260*x^2 +16617915*x-337500)*P^2+(69780960*x^2-37663785*x+506250)*P^3+(41957120*x-\ 47278080*x^2-337500)*P^4+(-23094720*x+11943936*x^2+84375)*P^5+5038848*x*P^6 = 0 The first, 30, terms, for the sake of the OEIS are [1, 5, 45, 520, 6925, 101106, 1574575, 25722540, 435980205, 7609063495, 135999566522, 2479263471540, 45955700857295, 864042041895440, 16446398097945960 , 316423498745778684, 6145707233548330605, 120369819323839533885, 2375294220535517061175, 47189110343871225768740, 943211115358145514966618, 18957271375509546161490300, 382940798013318995314035210, 7771271531645080780392016260, 158378140856039632273570280175, 3240395720619576323206879273761, 66538574976048232535123499088560, 1370907938168113059968323226988640, 28333548551355491869744280194081640, 587300397039122608225579805870404320] ---------------------------------------------------- This concludes this article for enuermationg Alpha(a,b) trees with a and b <=, 6 We were successful for the following pairs, {[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 0], [3, 1], [3, 2], [4, 0], [4, 2], [4, 3], [5, 2], [5, 3], [5, 4], [6, 5]} but failed for the following pairs, {[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [1, 6], [2, 3], [2, 4], [2, 5], [2, 6], [3, 3], [3, 4], [3, 5], [3, 6], [4, 1], [4, 4], [4, 5], [4, 6], [5, 0], [5, 1], [5, 5], [5, 6], [6, 0], [6, 1], [6, 2], [6, 3], [6, 4], [6, 6]} This colcludes this article, that took, 2811.118, seconds to generate.