OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
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)^(n+1))^(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)^(n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 2.2013296851505132606640400434738193121994558898350865326..., c = 0.026186121837027622395555466054900245177877028741031867... - Vaclav Kotesovec, Oct 06 2020, updated Feb 05 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 235*x^4 + 3470*x^5 + 61933*x^6 + 1274893*x^7 + 29423331*x^8 + 747440115*x^9 + 20636072811*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 4*x^2 + 31*x^3 + 356*x^4 + 5291*x^5 + 94592*x^6 + 1948763*x^7 + 45025516*x^8 + 1145651239*x^9 + 31696223593*x^10 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1, 2/2, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1, 3, 21, 235, 3470, 61933, 1274893, ...];
n=1: [1, 2, 7, 48, 521, 7536, 132657, 2704342, ...];
n=2: [1, 3, 12, 82, 867, 12288, 213282, 4304877, ...];
n=3: [1, 4, 18, 124, 1283, 17828, 305056, 6094832, ...];
n=4: [1, 5, 25, 175, 1780, 24271, 409380, 8094540, ...];
n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];
n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];
n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+1))^(n+1):
n=0: [1, 1, 1, 3, 21, 235, 3470, 61933, ...];
n=1: [1, 2, 5, 18, 114, 1166, 16355, 283142, ...];
n=2: [1, 3, 12, 55, 354, 3372, 44463, 739917, ...];
n=3: [1, 4, 22, 124, 857, 7908, 98244, 1558788, ...];
n=4: [1, 5, 35, 235, 1780, 16501, 195980, 2955095, ...];
n=5: [1, 6, 51, 398, 3321, 31746, 368032, 5294250, ...];
n=6: [1, 7, 70, 623, 5719, 57302, 662144, 9182013, ...];
n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...
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)^(m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 16 2018
STATUS
approved