A191802 - OEIS

%I #4 Mar 30 2012 18:37:26

%S 1,1,5,43,473,5942,81393,1186342,18132473,287948903,4722077279,

%T 79636530163,1377304530677,24382127678100,441294262119031,

%U 8160739579770316,154169018332135841,2975846752734820345,58718914018159811186

%N  G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(4*n^2).

%F  Let A = g.f. A(x), then A satisfies:

%F (1) A = Sum_{n>=0} x^n*A^(4*n)*Product_{k=1..n} (1-x*A^(16*k-12))/(1-x*A^(16*k-4));

%F (2) A = 1/(1- A^4*x/(1- A^4*(A^8-1)*x/(1- A^20*x/(1- A^12*(A^16-1)*x/(1- A^36*x/(1- A^20*(A^24-1)*x/(1- A^52*x/(1- A^28*(A^32-1)*x/(1- ...))))))))) (continued fraction);

%F due to a q-series identity and an identity of a partial elliptic theta function, respectively.

%e  G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 473*x^4 + 5942*x^5 + 81393*x^6 +...

%e where the g.f. satisfies:

%e A(x) = 1 + x*A(x)^4 + x^2*A(x)^16 + x^3*A(x)^36 + x^4*A(x)^64 +...+ x^n*A(x)^(4*n^2) +...

%o  (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(4*m^2)));polcoeff(A,n)}

%Y  Cf. A107595, A191800, A191801, A191803, A191804.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2011