OFFSET
0,2
FORMULA
Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*(1+A)^n*A^n * Product_{k=1..n} (1 - x*(1+A)*A^(2*k-1))/(1 - x*(1+A)*A^(2*k))
(2) A = 1/(1- A*(1+A)*x/(1- A*(A-1)*(1+A)*x/(1- A^3*(1+A)*x/(1- A^2*(A^2-1)*(1+A)*x/(1- A^5*(1+A)*x/(1- A^3*(A^3-1)*(1+A)*x/(1- A^7*(1+A)*x/(1- A^4*(A^4-1)*(1+A)*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 + 2*x + 10*x^2 + 74*x^3 + 658*x^4 + 6514*x^5 +...
Let A = g.f. A(x), then A satisfies:
A = 1 + x*(1+A)*A + x^2*(1+A)^2*A^3 + x^3*(1+A)^3*A^6 + x^4*(1+A)^4*A^10 +...
Equivalently,
A = 1 + x*(A + A^2) + x^2*(A^3 + 2*A^4 + A^5) + x^3*(A^6 + 3*A^7 + 3*A^8 + A^9) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1+A)^m*(A+x*O(x^n))^(m*(m+1)/2))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 26 2011
STATUS
approved