A167481 Convolution of the central binomial coefficients A000984(n) and (-2)^n.

1, 0, 6, 8, 54, 144, 636, 2160, 8550, 31520, 121716, 462000, 1780156, 6840288, 26436024, 102245472, 396589446, 1540427328, 5994280644, 23356702512, 91133123796, 355991626848, 1392115710024, 5449199307552, 21349205067996

Offset: 0

Author

Paul Barry, Nov 04 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A102591.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000984, A102591.

Programs

Mathematica

Table[FullSimplify[(-2)^n/Sqrt[3] + 1/2*Binomial[2*(1+n),1+n] * Hypergeometric2F1[1,3/2+n,2+n,-2]],{n,0,20}] (* Vaclav Kotesovec, Jan 31 2014 *)
CoefficientList[Series[1/((1 + 2*t)*Sqrt[1 - 4 t]), {t,0,50}], t] (* G. C. Greubel, Jun 13 2016 *)

Formula

G.f.: 1/((1+2x)*sqrt(1-4x)).

a(n) = Sum_{k=0..n} (-2)^(n-k)*C(2k,k).

Conjecture: n*a(n) + 2*(1-n)*a(n-1) + 4*(1-2n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011

a(n) = (-2)^n*JacobiP(n, 1/2, -1-n, -5). - Peter Luschny, Aug 02 2014