A215128 G.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / n!.

1, 2, 12, 64, 370, 2184, 13132, 79944, 491238, 3040400, 18926336, 118369368, 743199184, 4681668488, 29574616440, 187281906512, 1188494457492, 7556371963488, 48123031011036, 306929964849200, 1960230225450420, 12534313062502440, 80236414444623240

Offset: 0

Author

Paul D. Hanna, Aug 04 2012

Keywords

• nonn

Comments

Compare to: Sum_{n>=0} d^n/dx^n x^(2*n)/n! = 1/sqrt(1-4*x).

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 64*x^3 + 370*x^4 + 2184*x^5 + 13132*x^6 +...
such that, by definition:
A(x) = 1 + d/dx (x+x^2)^2 + d^2/dx^2 (x+x^2)^4/2! + d^3/dx^3 (x+x^2)^6/3! + d^4/dx^4 (x+x^2)^8/4! + d^5/dx^5 (x+x^2)^10/5! +...
		

Crossrefs

Cf. A214372, A215125, A215129, A038112.

Programs

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

a(n) = (n+1)*A214372(n+1), where G(x) = x + (G(x) + G(x)^2)^2 is the g.f. of A214372.