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%I #14 Oct 14 2020 11:19:23
%S 1,7,189,17283,2755053,604260531,165416203197,53736069429315,
%T 20098682471065149,8484270818691168963,3985069388942026022589,
%U 2060504358592580623699011,1162904612283296975554475517,711422819982429170172765550083,469007739834268780510389856367613,331521891387779056571085490125831171,250157485456407234540581483486760865533
%N G.f. A(x) satisfies: 4 = Sum_{n>=0} (3/4)^n * (1 + x)^(n^2) / A(x)^n.
%H Paul D. Hanna, <a href="/A303292/b303292.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: 4 = 1/(1 - q/(4/3*A(x) - q*(q^2-1)/(1 - q^5/(4/3*A(x) - q^3*(q^4-1)/(1 - q^9/(4/3*A(x) - q^5*(q^6-1)/(1 - q^13/(4/3*A(x) - q^7*(q^8-1)/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.
%F G.f.: 4 = Sum_{n>=0} (3/4)^n * (1+x)^n / A(x)^n * Product_{k=1..n} (4*A(x) - 3*(1+x)^(4*k-3)) / (4*A(x) - 3*(1+x)^(4*k-1)), due to a q-series identity.
%F a(n) ~ 3^(5/2) * 2^(2*n - 17/2) * n^n / (exp(n + log(4/3)^2 / 8) * log(4/3)^(2*n + 1)). - _Vaclav Kotesovec_, Oct 14 2020
%e G.f.: A(x) = 1 + 7*x + 189*x^2 + 17283*x^3 + 2755053*x^4 + 604260531*x^5 + 165416203197*x^6 + 53736069429315*x^7 + 20098682471065149*x^8 + ...
%e such that A = A(x) satisfies:
%e 4 = 1 + (1+x)/(4*A/3) + (1+x)^4/(4*A/3)^2 + (1+x)^9/(4*A/3)^3 + (1+x)^16/(4*A/3)^4 + (1+x)^25/(4*A/3)^5 + (1+x)^36/(4*A/3)^6 + (1+x)^49/(4*A/3)^7 + ...
%o (PARI) /* Find A(x) that satisfies the continued fraction: */
%o {a(n) = my(A=[1], q=1+x, CF=1); for(i=1, n, A=concat(A, 0); m=#A; for(k=0, m, CF = 1/(1 - q^(4*m-4*k+1)/(4/3*Ser(A) - q^(2*m-2*k+1)*(q^(2*m-2*k+2) - 1)*CF)) ); A[#A] = Vec(CF)[#A]/12 ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A303290, A303291.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 22 2018
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