A192401 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * x^n/(1 - x^(2*n)).

1, 1, 2, 6, 16, 49, 156, 512, 1728, 5959, 20886, 74204, 266624, 967141, 3536814, 13025478, 48266972, 179831935, 673258626, 2531481990, 9555606112, 36196916933, 137554950152, 524265889839, 2003513188296, 7675473295796, 29471911733772

Offset: 0

Author

Paul D. Hanna, Jun 30 2011

Keywords

• nonn

Comments

Related q-series identity:

Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=x, y=1, z=A(x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 156*x^6 +...
which satisfies the following relations:
A(x) = 1 + A(x)*x/(1-x^2) + A(x)^2*x^2/(1-x^4) + A(x)^3*x^3/(1-x^6) +...
A(x) = 1 + A(x)*x/(1-A(x)*x) + A(x)*x^3/(1-A(x)*x^3) + A(x)*x^5/(1-A(x)*x^5) +...
		

Crossrefs

Cf. A192400, A192403.

Programs

PARI

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

PARI

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

Formula

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