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A360336
G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n))^(n+1) for n >= 0.
3
1, 1, 6, 99, 2608, 90800, 3835458, 187727106, 10356030404, 632391914502, 42217751766193, 3053486035335835, 237640678130730437, 19794116975373467259, 1756875217029906875379, 165552614838271944281933, 16509692094523556884973416, 1737510282985845400007263814
OFFSET
0,3
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(3*n))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.0541176773983..., c = 0.03951220887392... - Vaclav Kotesovec, Feb 06 2023
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 99*x^3 + 2608*x^4 + 90800*x^5 + 3835458*x^6 + 187727106*x^7 + 10356030404*x^8 + 632391914502*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 7*x^2 + 118*x^3 + 3113*x^4 + 108221*x^5 + 4564720*x^6 + 223208259*x^7 + 12307249017*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(3*n))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 21/3, 472/4, 15565/5, 649326/6, 31953040/7, 1785666072/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1, 6, 99, 2608, 90800, 3835458, 187727106, ...];
n=1: [1, 2, 13, 210, 5450, 188004, 7893613, 384731112, ...];
n=2: [1, 3, 21, 334, 8544, 292017, 12186069, 591418401, ...];
n=3: [1, 4, 30, 472, 11909, 403268, 16725042, 808213780, ...];
n=4: [1, 5, 40, 625, 15565, 522211, 21523390, 1035561335, ...];
n=5: [1, 6, 51, 794, 19533, 649326, 26594644, 1273925322, ...];
n=6: [1, 7, 63, 980, 23835, 785120, 31953040, 1523791095, ...];
n=7: [1, 8, 76, 1184, 28494, 930128, 37613552, 1785666072, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(3*n))^(n+1):
n=0: [1, 1, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 2, 7, 48, 719, 17882, 603567, 25021464, ...];
n=2: [1, 3, 21, 190, 2814, 65460, 2105997, 84726534, ...];
n=3: [1, 4, 42, 472, 7303, 162828, 4982706, 193437168, ...];
n=4: [1, 5, 70, 940, 15565, 341796, 10002300, 373126910, ...];
n=5: [1, 6, 105, 1640, 29340, 649326, 18377374, 658075230, ...];
n=6: [1, 7, 147, 2618, 50729, 1150968, 31953040, 1101647800, ...];
n=7: [1, 8, 196, 3920, 82194, 1934296, 53433184, 1785666072, ...]; ...
to see that the main diagonals of the tables are the same.
PROG
(PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(3*m))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2023
STATUS
approved