%I #7 Mar 30 2012 18:37:27
%S 1,4,28,240,2348,24952,280192,3271232,39310668,483032980,6041149272,
%T 76648727632,984161689728,12764078032568,166969699620640,
%U 2200415358484800,29186416580736300,389340777798701672,5220028320540100220,70303231772070200912
%N G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/((1-x^n)*(1 - x^n*A(x)^2)).
%C Related q-series (Heine) identity:
%C 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)),
%C here q=x, x=x, y=z=A(x).
%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (A(x) + x^k)^2/(1-x^(k+1))^2 due to the Heine identity.
%e G.f.: A(x) = 1 + 4*x + 28*x^2 + 240*x^3 + 2348*x^4 + 24952*x^5 +...
%e The g.f. A = A(x) satisfies:
%e A = (1+x*A)^2/((1-x)*(1-x*A^2)) * (1+x^2*A)^2/((1-x^2)*(1-x^2*A^2)) * (1+x^3*A)^2/((1-x^3)*(1-x^3*A^2)) *...
%e A = {1 + x*(A+1)^2/(1-x)^2 + x^2*(A+1)^2*(A+x)^2/((1-x)*(1-x^2))^2 + x^3*(A+1)^2*(A+x)^2*(A+x^2)^2/((1-x)*(1-x^2)*(1-x^3))^2 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(k=1,n,(1+x^k*A)^2/((1-x^k+x*O(x^n))*(1-x^k*A^2))));polcoeff(A,n)}
%o (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)^2/(1-x^(k+1)+x*O(x^n))^2)));polcoeff(A,n)}
%Y Cf. A192624, A192620, A192622.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 06 2011