A128387 Expansion of c(5x^2)/(1-x*c(5x^2)), where c(x) is the g.f. of A000108.

1, 1, 6, 11, 66, 146, 876, 2131, 12786, 32966, 197796, 530526, 3183156, 8786436, 52718616, 148733571, 892401426, 2561439806, 15368638836, 44731364266, 268388185596, 790211926076, 4741271556456, 14095578557486

Offset: 0

Author

Paul Barry, Feb 28 2007

Keywords

• easy
• nonn

Comments

Hankel transform is 5^C(n+1,2).

Reversion of x*(1+x)/(1+2*x+6*x^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A009766, A120730, A126386.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-20*x^2)+2*x-1)/(2*x*(1-6*x)) )); // G. C. Greubel, Nov 07 2022

Mathematica

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
A126387[n_]:= Sum[5^k*A120730[n, n-k], {k,0,n}];
Table[A126387[n], {n, 0, 50}] (* G. C. Greubel, Nov 07 2022 *)

SageMath

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126387(n): return sum(5^k*A120730(n,n-k) for k in range(n+1))
[A126387(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

Formula

G.f.: (sqrt(1-20*x^2) + 2*x - 1)/(2*x*(1-6*x)).

a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^j*6^(k-j).

a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*5^k/(n-k+1).

a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*5^k.

a(n) = Sum_{k=0..n} 5^k*A120730(n,n-k). - Philippe Deléham, Mar 03 2007

(n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - R. J. Mathar, Nov 14 2011