A162533 a(n) = Sum_{k=0..n} binomial(n,2k)*A002426(k).

1, 1, 2, 4, 10, 26, 68, 176, 454, 1174, 3052, 7976, 20932, 55108, 145448, 384704, 1019462, 2706214, 7194956, 19155896, 51065260, 136284236, 364097912, 973654240, 2605983772, 6980545276, 18712478072, 50196568144, 134739960904, 361892443592, 972537193168

Offset: 0

Author

Paul Barry, Jul 05 2009

Keywords

• easy
• nonn

Comments

Hankel transform is (-1)^binomial(n,2)*(-2)^A128054(n) (see A128055).

Links

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

Crossrefs

Cf. A027826.

Programs

Mathematica

b[n_] := If[n < 0, 0, 3^n Hypergeometric2F1[1/2, -n, 1, 4/3]]; Table[Sum[Binomial[n, 2*k]*b[k], {k, 0, n}], {n, 0, 50}] (* or *) CoefficientList[Series[(1-x)/sqrt(1-4*x+4*x^2-4*x^4), {x, 0, 50}], x] (* G. C. Greubel, Feb 27 2017 *)

PARI

x='x+O('x^50); Vec((1-x)/sqrt(1-4*x+4*x^2-4*x^4)) \\ G. C. Greubel, Feb 27 2017

Formula

G.f.: (1-x)/((1-x)^2-x^2-2x^4/((1-x)^2-x^2-x^4/((1-x)^2-x^2-x^4/(1-... (continued fraction).

G.f.: (1-x)/sqrt(1-4*x+4*x^2-4*x^4) = (1-x)/sqrt((1-2*x)^2-4*x^4) = (1-x)/sqrt((1-x-2*x^2)*(1-x+2*x^2)). - Paul Barry, Oct 13 2009

Conjecture: n*a(n) + (4-5*n)*a(n-1) + 2*(4*n-7)*a(n-2) + 4*(3-n)*a(n-3) + 4*(2-n)*a(n-4) + 4*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 16 2011

a(n) ~ 3^(1/4) * (1 + sqrt(3))^(n + 1/2) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jun 08 2019