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

0,3

Comments

Hankel transform is A102591.

Links

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

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

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 *)

Crossrefs

Cf. A000984, A102591.

Sequence in context: A223852 A227167 A192158 * A137122 A272963 A218215

Adjacent sequences: A167478 A167479 A167480 * A167482 A167483 A167484

Keyword

easy,nonn

Author

Paul Barry, Nov 04 2009

Status

approved