OFFSET
0,3
FORMULA
The g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(2*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^2)/(1 + x^(4*k-1)*A(x)^2);
(2) 1 = A(x)/(1+ x*A(x)^2/(1- x*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1-x^4)*A(x)^2/(1+ x^9*A(x)^2/(1- x^5*(1-x^6)*A(x)^2/(1+ x^13*A(x)^2/(1- x^7*(1-x^8)*A(x)^2/(1- ...))))))))) (continued fraction);
due to identities of a partial elliptic theta function.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 265*x^5 + 1373*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 + x^4*A(x)^5 - x^9*A(x)^7 + x^16*A(x)^9 -+...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 270*x^4 + 1398*x^5 + 7518*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 130*x^3 + 695*x^4 + 3816*x^5 +...
PROG
(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(2*m+1) ), #A-1)); if(n<0, 0, A[n+1])}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 16 2011
STATUS
approved