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A192625 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/((1-x^n)*(1 - x^n*A(x)^2)).

%I A192625 #13 Jun 30 2025 06:55:54
%S A192625 1,4,28,240,2348,24952,280192,3271232,39310668,483032980,6041149272,
%T A192625 76648727632,984161689728,12764078032568,166969699620640,
%U A192625 2200415358484800,29186416580736300,389340777798701672,5220028320540100220,70303231772070200912
%N A192625 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/((1-x^n)*(1 - x^n*A(x)^2)).
%C A192625 Related q-series (Heine) identity:
%C A192625 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 A192625 here q=x, x=x, y=z=A(x).
%H A192625 Vaclav Kotesovec, <a href="/A192625/b192625.txt">Table of n, a(n) for n = 0..300</a>
%F A192625 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.
%F A192625 a(n) ~ c * d^n / n^(3/2), where d = 14.589835921559349866989401284706614286378779559... and c = 0.453836352355009937995115064134624562210185... - _Vaclav Kotesovec_, Jun 30 2025
%e A192625 G.f.: A(x) = 1 + 4*x + 28*x^2 + 240*x^3 + 2348*x^4 + 24952*x^5 +...
%e A192625 The g.f. A = A(x) satisfies:
%e A192625 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 A192625 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 +...
%t A192625 (* Calculation of constants {d,c}: *) Chop[{1/r, (1/(2*Sqrt[Pi]))*s*(-1 + s^2)* Sqrt[(Log[ r]*(r*Log[r]*QPochhammer[-s, r]*QPochhammer[s^2, r] * Derivative[0, 1][QPochhammer][r, r] + QPochhammer[r, r]*(-2*r*Log[r]*QPochhammer[s^2, r]* Derivative[0, 1][QPochhammer][-s, r] + QPochhammer[-s, r]*((-QPochhammer[s^2, r])*(Log[1 - r] + QPolyGamma[0, 1, r]) + r*Log[r] * Derivative[0, 1][QPochhammer][s^2, r]))))/(QPochhammer[ r, r]*QPochhammer[-s, r]*QPochhammer[s^2, r] * (s*(1 + s^2) * Log[r]^2 + (-1 + s^2)^2 * QPolyGamma[1, Log[-s]/Log[r], r] - 2*(-1 + s^2)^2 * QPolyGamma[1, (2*Log[s])/Log[r], r]))]} /. FindRoot[{s + ((-1 + s) * QPochhammer[-s, r]^2)/((1 + s) * QPochhammer[r, r] * QPochhammer[s^2, r]) == 0, (2*s)/(-1 + s^2) + (2*(-QPolyGamma[0, Log[-s]/Log[r], r] + QPolyGamma[0, (2*Log[s])/Log[r], r]))/Log[r] == 1}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120]] (* _Vaclav Kotesovec_, Jun 30 2025 *)
%o A192625 (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 A192625 (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 A192625 Cf. A192624, A192620, A192622.
%K A192625 nonn
%O A192625 0,2
%A A192625 _Paul D. Hanna_, Jul 06 2011