A215129 G.f.: Sum_{n>=0} d^n/dx^n (2*x + x^2)^(2*n) / n!.

1, 8, 108, 1604, 25000, 400680, 6539932, 108124704, 1804745052, 30345973020, 513246523504, 8722051475280, 148808353519644, 2547317293116512, 43729600850054640, 752553083643333008, 12978731348733064680, 224258766666165861192

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 (2*x+x^2)^2 + d^2/dx^2 (2*x+x^2)^4/2! + d^3/dx^3 (2*x+x^2)^6/3! + d^4/dx^4 (2*x+x^2)^8/4! + d^5/dx^5 (2*x+x^2)^10/5! +...
		

Crossrefs

Cf. A215128.

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, (2*x+x^2+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f.: A(x) = d/dx G(x), where G(x) = x + (2*G(x) + G(x)^2)^2.