OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 4.01604513838270620496843653760987690323... and c = 2.07544072297996637757124624302382219... - Vaclav Kotesovec, Sep 27 2023
Radius of convergence r = 0.2490011853807768883971843288180859269 = 1/d and A(r) = 3.261386924996517219078267128734843819... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 + r^n*A(r)) and (2) A(r) = Product_{n>=1} (1 + r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024
EXAMPLE
G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 31*x^4 + 102*x^5 + 342*x^6 + 1167*x^7 + 4046*x^8 + 14213*x^9 + 50464*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2 * (1 + x^3*A(x))^3 * (1 + x^4*A(x))^4 * ...
MATHEMATICA
nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 04 2018
STATUS
approved