A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.

1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 63, 117, 211, 383, 713, 1348, 2547, 4793, 9039, 17165, 32785, 62761, 120243, 230768, 444119, 857015, 1656931, 3207990, 6219994, 12079544, 23496417, 45767352, 89256038, 174269488, 340646238, 666604642

Offset: 0

Author

Paul D. Hanna, Feb 24 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 10*x^3 + 18*x^4 + 36*x^5 + 73*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 80*x^5 + 168*x^6 +...
where
A(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 +...
		

Crossrefs

Cf. A157135, A157133, A157136.

Cf. A107595. [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(n),x^(m^2)*A^m)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = g.f. of A157135,

where A157135(n) = [x^n] A(x)^(n+1)/(n+1) for n>=0,

and a(n) = [x^n] -1/B(x)^(n-1)/(n-1) for n>1.

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^3-x)*A(x)/(1- x^5*A(x)/(1- (x^7-x^3)*A(x)/(1- x^9*A(x)/(1- (x^11-x^5)*A(x)/(1- x^13*A(x)/(1- (x^15-x^7)*A(x)/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

(End)

From Paul D. Hanna, May 05 2010: (Start)

Let A = g.f. A(x) at x=q, then A satisfies the q-series:

A = Sum_{n>=0} q^n*A^n*Product_{k=1..n} (1-q^(4k-3)*A)/(1-q^(4k-1)*A).

(End)

Extensions

Typo in data corrected by D. S. McNeil, Aug 17 2010