OFFSET
0,2
FORMULA
Let A = g.f. A(x), then A satisfies:
(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))
(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).
The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.
EXAMPLE
G.f.: A(x) = 1 + 4*x + 48*x^2 + 912*x^3 + 21184*x^4 + 552320*x^5 +...
Let A = g.f. A(x), then A satisfies:
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 +...
Equivalently,
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) +...
PROG
(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)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 26 2011
STATUS
approved