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A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^n)^2/(1 - x^(n+1)*A(x)^n)^2.

%I A192622 #14 Jun 30 2025 07:17:50
%S A192622 1,4,12,48,220,1080,5600,30112,166300,937620,5374200,31221488,
%T A192622 183430656,1087975256,6505878592,39179738400,237412139260,
%U A192622 1446488046824,8855937880108,54455375407504,336159421649528,2082508824181856,12942736191473792
%N A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^n)^2/(1 - x^(n+1)*A(x)^n)^2.
%C A192622 Related q-series (Heine) identity:
%C A192622 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.
%H A192622 Vaclav Kotesovec, <a href="/A192622/b192622.txt">Table of n, a(n) for n = 0..160</a>
%F A192622 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.
%F A192622 Self-convolution of A192623.
%F A192622 a(n) ~ c * d^n / n^(3/2), where d = 6.65133046938958271... and c = 1.095759838870545... - _Vaclav Kotesovec_, Jun 30 2025
%e A192622 G.f.: A(x) = 1 + 4*x + 12*x^2 + 48*x^3 + 220*x^4 + 1080*x^5 +...
%e A192622 The g.f. A = A(x) satisfies the following relations:
%e A192622 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 *...
%e A192622 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)) +...
%t A192622 (* 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 *)
%o A192622 (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)}
%o A192622 (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)}
%Y A192622 Cf. A192623, A192620, A192624.
%K A192622 nonn
%O A192622 0,2
%A A192622 _Paul D. Hanna_, Jul 06 2011