A166588 Partial sums of A097331; binomial transform of A166587.

1, 2, 2, 3, 3, 5, 5, 10, 10, 24, 24, 66, 66, 198, 198, 627, 627, 2057, 2057, 6919, 6919, 23715, 23715, 82501, 82501, 290513, 290513, 1033413, 1033413, 3707853, 3707853, 13402698, 13402698, 48760368, 48760368, 178405158, 178405158, 656043858

Offset: 0

Author

Paul Barry, Oct 17 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A131713. The Hankel transform of the sequence 1,1,2,2,... is A128017(n+3). A155587 doubled.

Links

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

Programs

Mathematica

CoefficientList[Series[(1+2*x-Sqrt[1-4*x^2])/(2*x*(1-x)), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

G.f.: (1+2x-sqrt(1-4x^2))/(2x(1-x))=((1+x^2*c(x^2))/(1-x)-1)/x, c(x) the g.f. of A000108.

a(n) = Sum_{k=0..n} C(n,k)*A166587(k).

Conjecture: (-n-1)*a(n) + (n+1)*a(n-1) + 4*(n-2)*a(n-2) + 4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012

a(n) ~ 2^(n+1/2) * (3-(-1)^n) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014