OFFSET
0,3
FORMULA
Let A = g.f. A(x), then A satisfies:
(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));
(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);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 473*x^4 + 5942*x^5 + 81393*x^6 +...
where the g.f. satisfies:
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) +...
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2011
STATUS
approved