OFFSET
0,2
COMMENTS
Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x*A(x), x=x, y=z=1.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..160
FORMULA
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (1 + x^k*A(x)^k)^2/((1 - x^(k+1)*A(x)^k)*(1 - x^(k+1)*A(x)^(k+1)) due to the Heine identity.
Self-convolution of A192623.
a(n) ~ c * d^n / n^(3/2), where d = 6.65133046938958271... and c = 1.095759838870545... - Vaclav Kotesovec, Jun 30 2025
EXAMPLE
G.f.: A(x) = 1 + 4*x + 12*x^2 + 48*x^3 + 220*x^4 + 1080*x^5 +...
The g.f. A = A(x) satisfies the following relations:
A = (1+x)^2/(1-x)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A^2)^2/(1-x^3*A^2)^2 *...
A = 1 + 4*x/((1-x)*(1-x*A)) + 4*x^2*(1+x*A)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)) + 4*x^3*(1+x*A)^2*(1+x^2*A^2)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x^3*A^3)) +...
MATHEMATICA
(* Calculation of constants {d, c}: *) Chop[{1/r, s*Sqrt[(QPochhammer[r, r*s]*(Log[r*s] - 2*QPolyGamma[0, Log[-r]/Log[r*s], r*s] + 2*QPolyGamma[0, Log[r]/Log[r*s], r*s]))/(Pi* Log[r*s]*(QPochhammer[r, r*s] - 4*r*s*(Derivative[0, 1][QPochhammer][r, r*s] - r*Sqrt[s]*Derivative[0, 2][QPochhammer][-r, r*s] + r*s*Derivative[0, 2][QPochhammer][r, r*s])))]} /. FindRoot[{QPochhammer[-r, r*s]^2/QPochhammer[r, r*s]^2 == s, QPochhammer[r, r*s]^2 + 2*r*s*QPochhammer[r, r*s] * Derivative[0, 1][QPochhammer][r, r*s] == 2*r*QPochhammer[-r, r*s] * Derivative[0, 1][QPochhammer][-r, r*s]}, {r, 1/6}, {s, 3}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(k=0, n, (1+x^(k+1)*A^k)^2/(1-x^(k+1)*(A+x*O(x^n))^k)^2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*prod(k=0, m-1, (1+x^k*A^k)^2/((1-x^(k+1)*A^k +x*O(x^n))*(1-x^(k+1)*A^(k+1)))))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 06 2011
STATUS
approved