A157133 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * A(x)^n.

1, 1, 1, 2, 4, 7, 14, 30, 62, 129, 278, 604, 1313, 2883, 6386, 14203, 31733, 71272, 160725, 363670, 825653, 1880351, 4293985, 9830499, 22558939, 51880565, 119552907, 276012657, 638348123, 1478749229, 3430799333, 7971134523

Offset: 0

Author

Paul D. Hanna, Feb 24 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 14*x^6 + 30*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 26*x^5 + 54*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 30*x^4 + 66*x^5 + 145*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 59*x^4 + 140*x^5 + 326*x^6 +...
where
A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 +...
		

Crossrefs

Cf. A157134, A157135, A157136.

Cf. A121690. [From Paul D. Hanna, Apr 25 2010]

Programs

PARI

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

Formula

Contribution from Paul D. Hanna, Apr 25 2010: (Start)

G.f. A(x) satisfies the continued fraction:

A(x) = 1/(1- x*A(x)/(1- (x^2-x)*A(x)/(1- x^3*A(x)/(1- (x^4-x^2)*A(x)/(1- x^5*A(x)/(1- (x^6-x^3)*A(x)/(1- x^7*A(x)/(1- (x^8-x^4)*A(x)/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

(End)