A176719 G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*x^n * A(x)^(n^2).

1, 2, 6, 30, 194, 1442, 11782, 103102, 951554, 9173186, 91780614, 948985822, 10110931650, 110794764642, 1247186300934, 14412811665278, 170949340705794, 2081185257723778, 26012832364535814, 333929563132811678, 4404347475363755714, 59705917899701420834, 832080588205468939782

Offset: 0

Author

Paul D. Hanna, Apr 25 2010

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 30*x^3 + 194*x^4 + 1442*x^5 +...
A(x) = 1 + 2*x*A(x) + 2*x^2*A(x)^4 + 2*x^3*A(x)^9 + 2*x^4*A(x)^16 + ...
Contribution from _Paul D. Hanna_, May 11 2010: (Start)
Given g.f. A(x), then Q = A(-x^2) satisfies the q-series:
Q/(1-x) = 1 + x*(xQ;Q)_1/(-xQ;Q)_1 + x^2*(xQ;Q)_2/(-xQ;Q)_2 +...
where the q-Pochhammer symbol (x;q)_n = Product_{k=0..n-1} (1-x*q^k). (End)
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A107595, A176720.

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))^(m^2)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

Contribution from Paul D. Hanna, May 11 2010: (Start)

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^k)/(1 + x*Q^k)

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

Extensions

Edited by Paul D. Hanna, Apr 27 2010