OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..250
FORMULA
Self-convolution of A190862.
a(n) ~ c * d^n / n^(3/2), where d = 6.693428901114353533300254329706934045134... and c = 4.7342954578062245798237099751798009... - Vaclav Kotesovec, Jun 30 2025
EXAMPLE
G.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 556*x^4 + 3048*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x*A^(1/2))^2/(1-x*A^(1/2))^2 * (1+x^2*A^(1/2))^2/(1-x^2*A^(1/2))^2 * (1+x^3*A^(1/2))^2/(1-x^3*A^(1/2))^2 *...
MATHEMATICA
(* Calculation of constants {d, c}: *) Chop[{1/r, (1 - s)*s*Log[r]* Sqrt[(2*r*(QPochhammer[Sqrt[s], r]* Derivative[0, 1][QPochhammer][-Sqrt[s], r] - QPochhammer[-Sqrt[s], r]* Derivative[0, 1][QPochhammer][Sqrt[s], r]))/ (Pi* QPochhammer[s, r^2]*(-2*Sqrt[s]*(1 + s)*Log[r]^2 + (-1 + s)^2* QPolyGamma[1, Log[s]/Log[r^2], r] - (-1 + s)^2* QPolyGamma[1, (2*I*Pi + Log[s])/Log[r^2], r]))]} /. FindRoot[{((-1 + Sqrt[s])^2* QPochhammer[-Sqrt[s], r]^2)/((1 + Sqrt[s])^2* QPochhammer[Sqrt[s], r]^2) == s, (-1 - 2*Sqrt[s] + s)/(-1 + s) + (QPolyGamma[0, Log[-Sqrt[s]]/Log[r], r] - QPolyGamma[0, Log[s]/(2*Log[r]), r])/ Log[r] == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+x^m*A^(1/2))^2/(1-x^m*A^(1/2)+x*O(x^n))^2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+sum(m=1, n, x^m*A^(m/2)*prod(k=1, m, (1+x^(k-1))/(1-x^k+x*O(x^n)))))^2); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 06 2011
STATUS
approved