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%I #12 Jan 19 2024 07:30:38
%S 1,2,4,10,36,136,548,2316,10050,44426,199666,910090,4196984,19545844,
%T 91791112,434181656,2066656564,9891669820,47578282002,229858639366,
%U 1114895656402,5427058308018,26503888167186,129821343271168,637626106479490
%N G.f. satisfies: A(x) = 1 + x*f(x, A(x)) where f(,) is Ramanujan's two-variable theta function.
%C Ramanujan's two-variable theta function is defined by:
%C f(a,b) = Sum_{n=-infinity..+infinity} a^(n*(n+1)/2) * b^(n*(n-1)/2).
%H Vaclav Kotesovec, <a href="/A192502/b192502.txt">Table of n, a(n) for n = 0..400</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions.</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions.</a>
%F G.f. satisfies:
%F (1) A(x) = 1+x + x*Sum_{n>=1} (x*A(x))^(n*(n-1)/2) * (x^n + A(x)^n).
%F (2) A(x) = 1 + x*Product_{n>=0} (1+x*q^n)*(1+A(x)*q^n)*(1-q^(n+1)) where q=x*A(x), due to Jacobi's triple product identity.
%F a(n) ~ c * d^n / n^(3/2), where d = 5.2286591857647664516287778... and c = 0.4431871616898705063582... - _Vaclav Kotesovec_, Sep 04 2017
%F Formula (2) can be rewritten as the functional equation y = 1 + x*QPochhammer(-x, x*y) * QPochhammer(-y, x*y) * QPochhammer(x*y). - _Vaclav Kotesovec_, Jan 19 2024
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 36*x^4 + 136*x^5 + 548*x^6 +...
%e The g.f. A = A(x) satisfies:
%e (1) A = 1+x + x*[(x+A) + x*A*(x^2+A^2) + x^3*A^3*(x^3+A^3) + x^6*A^6*(x^4+A^4) + x^10*A^10*(x^5+A^5) +...].
%e (2) A = 1 + x*(1+x)*(1+A)*(1-x*A)* (1+x^2*A)*(1+x*A^2)*(1-x^2*A^2)* (1+x^3*A^2)*(1+x^2*A^3)*(1-x^3*A^3)* (1+x^4*A^3)*(1+x^3*A^4)*(1-x^4*A^4)*...
%t (* Calculation of constant d: *) 1/r /. FindRoot[{s == 1 + r*QPochhammer[-r, r*s] * QPochhammer[-s, r*s] * QPochhammer[r*s], r*(-1 + s) * Derivative[0, 1][QPochhammer][-r, r*s] / QPochhammer[-r, r*s] + r^2*QPochhammer[-r, r*s] * QPochhammer[r*s] * Derivative[0, 1][QPochhammer][-s, r*s] + (-1 + s)*(-((2*Log[1 - r*s] + QPolyGamma[0, 1, r*s] + QPolyGamma[0, Log[-s]/Log[r*s], r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s] / QPochhammer[r*s]) == 1}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 19 2024 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x+x*sum(m=1,sqrtint(2*n)+1,(x*A+x*O(x^n))^(m*(m-1)/2)*(x^m+A^m)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,q=x*(A+O(x^n));A=1+x*prod(m=0,n,(1+x*q^m)*(1+A*q^m)*(1-q^(m+1))) );polcoeff(A,n)}
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 03 2011
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