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

0,3

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

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

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 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-20*x^2)+2*x-1)/(2*x*(1-6*x)) )); // 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

Crossrefs

Cf. A000108, A009766, A120730, A126386.

Sequence in context: A365496 A110445 A305608 * A061519 A193664 A080875

Adjacent sequences: A128384 A128385 A128386 * A128388 A128389 A128390

Keyword

easy,nonn

Author

Paul Barry, Feb 28 2007

Status

approved