A127362 a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-3)^(n-k).

1, -2, 8, -24, 84, -272, 920, -3040, 10180, -33840, 112968, -376224, 1254696, -4181088, 13939248, -46459584, 154873860, -516229040, 1720795880, -5735921440, 19119861304, -63732624672, 212442552528, -708140901184, 2360471473384, -7868234639072

Offset: 0

Author

Paul Barry, Jan 11 2007

Keywords

• easy
• sign

Comments

Hankel transform is 4^n. In general, for r>=0, the sequence given by sum{k=0..n, C(n,floor(k/2))*(-r)^(n-k)} has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+3x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Mathematica

CoefficientList[Series[1/(-1+2*x+2*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1+3*x*c(x^2)).

D-finite with recurrence 3*n*a(n) +2*(5*n-3)*a(n-1) +4*(-3*n+1)*a(n-2) +40*(-n+2)*a(n-3)=0. - R. J. Mathar, Nov 15 2012

a(n) ~ (-1)^n * 2^(n+1) * 5^n / 3^(n+1). - Vaclav Kotesovec, Feb 08 2014

G.f.: 1/(-1+2*x+2*sqrt(1-4*x^2)). - Vaclav Kotesovec, Feb 08 2014