A133443 a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-1)^k*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, 26227455730640

Offset: 0

Author

Philippe Deléham, Nov 26 2007, Dec 07 2007

Keywords

• nonn

Comments

Hankel transform is 4^n. Second binomial transform is A076035.

Links

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

Crossrefs

Cf. A000108, A015518, A053121, A076035, A127362.

Programs

Mathematica

Table[Sum[Binomial[n,Floor[k/2]]*(-1)^k*3^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

Formula

a(n) = Sum_{k=0..n} A053121(n,k)*A015518(k+1) = (-1)^n*A127362(n). G.f.: (1/sqrt(1-4*x^2))*(1-x*c(x^2))/(1-3*x*c(x^2)), where c(x) is the g.f. of Catalan numbers A000108.

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). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2*10^n/3^(n+1). - Vaclav Kotesovec, Oct 20 2012

Extensions

More terms from Vincenzo Librandi, May 25 2013