A192624 G.f. satisfies: A(x) = Product_{n>=1} (1+x^n)*(1 + x^n*A(x))/((1-x^n)*(1 - x^n*A(x))).

1, 4, 20, 112, 676, 4328, 28912, 199392, 1409364, 10157828, 74375640, 551715264, 4137527408, 31318286632, 238958947328, 1835960454272, 14192132860868, 110298595778872, 861338925309604, 6755283201399776, 53185599585579640

Offset: 0

Author

Paul D. Hanna, Jul 06 2011

Keywords

• nonn

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, x=x, y=A(x), z=1.

Examples

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4328*x^5 +...
The g.f. A = A(x) satisfies:
A = (1+x)*(1+x*A)/((1-x)*(1-x*A)) * (1+x^2)*(1+x^2*A)/((1-x^2)*(1-x^2*A)) * (1+x^3)*(1+x^3*A)/((1-x^3)*(1-x^3*A)) *...
A = {1 + 2*x*(A+1)/(1-x)^2 + 2*x^2*(A+1)*(A+x)*(1+x)/((1-x)*(1-x^2))^2 + 2*x^3*(A+1)*(A+x)*(A+x^2)*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3))^2 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Crossrefs

Cf. A192625, A192620, A192622.

Programs

Mathematica

(* Calculation of constants {d,c}: *) Chop[{1/r, (1/(2*Sqrt[Pi]))* Sqrt[((-1 + s)^2*s*(1 + s)* Log[r]*(-2*r*s*(1 + s)*Log[r]*QPochhammer[r, r]*QPochhammer[s, r]* Derivative[0, 1][QPochhammer][-1, r] + r*(-1 + s)*Log[r]*QPochhammer[-1, r]^2* Derivative[0, 1][QPochhammer][-s, r] + 2*s*(1 + s)*QPochhammer[-1, r]* (r*Log[r]*QPochhammer[s, r]* Derivative[0, 1][QPochhammer][r, r] + QPochhammer[r, r]* ((-QPochhammer[s, r])*(Log[1 - r] + QPolyGamma[0, 1, r]) + r*Log[r]*Derivative[0, 1][QPochhammer][s, r]))))/ (QPochhammer[-1, r]*QPochhammer[r, r]* QPochhammer[s, r]*(2*s*(1 + s^2)*Log[r]^2 + (-1 + s^2)^2* QPolyGamma[1, Log[-s]/Log[r], r] - (-1 + s^2)^2* QPolyGamma[1, Log[s]/Log[r], r]))]} /. FindRoot[{2* s + ((-1 + s)*QPochhammer[-1, r]*QPochhammer[-s, r])/((1 + s)* QPochhammer[r, r]*QPochhammer[s, r]) == 0, (2*s)/(-1 + s^2) + (-QPolyGamma[0, Log[-s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r])/Log[r] == 1}, {r, 1/8}, {s, 3}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)

PARI

{a(n)=local(A=1+x);for(i=1,n,A=prod(k=1,n,(1+x^k*A)*(1+x^k)/((1-x^k+x*O(x^n))*(1-x^k*A))));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,(A+x^k)*(1+x^k)/(1-x^(k+1)+x*O(x^n))^2)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (A(x) + x^k)*(1+x^k)/(1-x^(k+1))^2 due to the Heine identity.

a(n) ~ c * d^n / n^(3/2), where d = 8.48061441094248725740275431395... and c = 1.3507361818605928928927572... - Vaclav Kotesovec, Jun 30 2025