OFFSET
0,3
FORMULA
G.f. A(x) satisfies the continued fraction:
1 = A(x)/(1+ x*A(x)^2/(1- x*(1+x)*A(x)^2/(1+ x^3*A(x)^2/(1+ x^2*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1+x^3)*A(x)^2/(1+ x^7*A(x)^2/(1+ x^4*(1-x^4)*A(x)^2/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 328*x^5 + 1796*x^6 +...
which satisfies:
1 = A(x) - x*A(x)^3 - x^3*A(x)^5 + x^6*A(x)^7 + x^10*A(x)^9 - x^15*A(x)^11 - x^21*A(x)^13 ++--...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 303*x^4 + 1662*x^5 + 9447*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 760*x^4 + 4401*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(2*(#A))+1, (-x)^(m*(m+1)/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