%I #10 Jun 20 2016 22:46:24
%S 1,4,48,912,21184,552320,15532032,460947712,14247537664,454761822208,
%T 14902431522816,499315007266816,17054726818791424,592541668923539456,
%U 20907267781281054720,748286964823747526656,27143591551031801806848,27143591551031801806848,997356616630147913089024,37108619649604340227768320,1397931208210552892111716352,53322215792785853528148017152,2059866344459108561028558880768,80619871370319975775336625340416
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^(2*n) * A(x)^(n^2).
%F Let A = g.f. A(x), then A satisfies:
%F (1) A = Sum_{n>=0} x^n*(1+A)^(2*n)*A^n * Product_{k=1..n} (1 - x*(1+A)^2*A^(4*k-3))/(1 - x*(1+A)^2*A^(4*k-1))
%F (2) A = 1/(1- A*(1+A)^2*x/(1- A*(A^2-1)*(1+A)^2*x/(1- A^5*(1+A)^2*x/(1- A^3*(A^4-1)*(1+A)^2*x/(1- A^9*(1+A)^2*x/(1- A^5*(A^6-1)*(1+A)^2*x/(1- A^13*(1+A)^2*x/(1- A^7*(A^8-1)*(1+A)^2*x/(1- ...))))))))) (continued fraction).
%F The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.
%e G.f.: A(x) = 1 + 4*x + 48*x^2 + 912*x^3 + 21184*x^4 + 552320*x^5 +...
%e Let A = g.f. A(x), then A satisfies:
%e A = 1 + x*(1+A)^2*A + x^2*(1+A)^4*A^4 + x^3*(1+A)^6*A^9 + x^4*(1+A)^8*A^16 +...
%e Equivalently,
%e A = 1 + x*(A + 2*A^2 + A^3) + x^2*(A^4 + 4*A^5 + 6*A^6 + 4*A^7 + A^8) + x^3*(A^9 + 6*A^10 + 15*A^11 + 20*A^12 + 15*A^13 + 6*A^14 + A^15) +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(1+A)^(2*m)*(A+x*O(x^n))^(m^2)));polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A107595, A192259.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 26 2011