A218294 G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*x^n * A(x)^(2*n^2).

1, 2, 10, 82, 866, 10482, 138698, 1957346, 29024642, 448005922, 7153738058, 117681081522, 1988787934818, 34465473701522, 611806834645642, 11118408274591938, 206835953956603394, 3939803761941599042, 76880490874588995978, 1538019374456939130386

Offset: 0

Author

Paul D. Hanna, Oct 26 2012

Keywords

• nonn

Comments

Given g.f. A(x), then Q = A(-x^2) satisfies:

Q = (1-x)*Sum_{n>=0} x^n*Product_{k=1..n} (1 - x*Q^(2*k))/(1 + x*Q^(2*k))

due to a q-series expansion for the Jacobi theta_4 function.

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 82*x^3 + 866*x^4 + 10482*x^5 + 138698*x^6 +...
where
A(x) = 1 + 2*x*A(x)^2 + 2*x^2*A(x)^8 + 2*x^3*A(x)^18 + 2*x^4*A(x)^32 + ...
		

Crossrefs

Cf. A176719, A218295.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 2*x^m*(A+x*O(x^n))^(2*m^2))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))