A192403 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * 2*x^n/(1 - 2*x^(2*n)).

1, 2, 6, 26, 106, 474, 2210, 10638, 52578, 265286, 1360702, 7074030, 37191694, 197398394, 1056255758, 5691813546, 30860701490, 168236407482, 921576598970, 5070138584230, 28002574339634, 155204886300414, 862985636296302, 4812513873922710

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=2, z=A(x).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 106*x^4 + 474*x^5 + 2210*x^6 +...
which satisfies the following relations:
A(x) = 1 + A(x)*2*x/(1-2*x^2) + A(x)^2*2*x^2/(1-2*x^4) + A(x)^3*2*x^3/(1-2*x^6) +...
A(x) = 1 + 2*A(x)*x/(1-A(x)*x) + 4*A(x)*x^3/(1-A(x)*x^3) + 8*A(x)*x^5/(1-A(x)*x^5) +...
		

Crossrefs

Cf. A192401, A192402.

Programs

PARI

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