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%I #20 Mar 03 2024 09:56:45
%S 1,2,12,88,726,6456,60392,585792,5838764,59440250,615431464,
%T 6460681656,68607630680,735682014648,7954732578032,86635206695808,
%U 949518438959574,10464751843723840,115904823140622164,1289419736206548408,14401729960605163272
%N G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^2)/(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)); here q=x, x=x*A(x)^2, y=1, z=0.
%H Vaclav Kotesovec, <a href="/A192621/b192621.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.
%F (2) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x)^2)*(1-x^k)), due to the Heine identity.
%F (3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x)^2)*(1-x^k), due to the Heine identity.
%F Self-convolution yields A192620.
%F a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 0.323075847195701225672585138139173170517867693... - _Vaclav Kotesovec_, Oct 04 2023
%F Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 1.624363189835514855585723923742556266289... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^4) ) and A(r) = Product_{n>=1} (1 + r^n*A(r)^2)/(1 - r^n*A(r)^2). - _Paul D. Hanna_, Mar 02 2024
%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 88*x^3 + 726*x^4 + 6456*x^5 + ...
%e The g.f. A = A(x) satisfies the following relations:
%e (0) A = (1+x*A^2)/(1-x*A^2) * (1+x^2*A^2)/(1-x^2*A^2) * (1+x^3*A^2)/(1-x^3*A^2) * ...
%e (1) A = 1 + 2*x*A^2/(1-x) + 2*x^2*A^4*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^6*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...
%e (2) A = 1 + 2*x*A^2/((1-x*A^2)*(1-x)) + 2*x^3*A^4*(1+x)/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 2*x^6*A^6*(1+x)*(1+x^2)/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...
%e (3) A^2 = 1 + 4*x*A^2/((1-x*A^2)*(1-x)) + 4*x^2*A^4*(1+x)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 4*x^3*A^6*(1+x)^2*(1+x^2)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...
%t nmax = 30; A[_] = 0; Do[A[x_] = Product[(1 + x^k*A[x]^2)/(1 - x^k*A[x]^2), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Oct 04 2023 *)
%t (* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-s^2, r] / QPochhammer[s^2, r] == s*((1 + s^2)/(1 - s^2)), QPolyGamma[0, Log[-s^2]/Log[r], r] - QPolyGamma[0, Log[s^2]/Log[r], r] == (2*(s^2/(s^4 - 1)) - 1/2) * Log[r]}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Oct 04 2023 *)
%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*A^2+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sqrt(1+sum(m=1,n,x^m*A^(2*m)*prod(k=1,m,(1+x^(k-1))^2/((1-x^k*A^2 +x*O(x^n))*(1-x^k))))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^(m*(m+1)/2)*A^(2*m)*prod(k=1,m,(1+x^(k-1))/((1-x^k*A^2 +x*O(x^n))*(1-x^k)))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^(2*m)*prod(k=1,m,(1+x^(k-1))/(1-x^k+x*O(x^n)))) );polcoeff(A,n)}
%Y Cf. A192620 (g.f. A(x)^2), A192623, A190862, A196150, A196151.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 06 2011
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