A133915 a(n) = Sum_{i=0..n} C(2*n-i,n+i)*2^i.

1, 2, 8, 30, 116, 452, 1772, 6974, 27524, 108852, 431168, 1709996, 6788536, 26971856, 107235668, 426594110, 1697855876, 6760326116, 26927208368, 107288242820, 427596003416, 1704598377176, 6796820059928, 27106584400460, 108123625907816, 431355955330952

Offset: 0

Author

Paul Barry, Sep 28 2007

Keywords

• easy
• nonn

Comments

A transform of the Jacobsthal numbers A001045(n+1) under the mapping g(x)->(1/(c(x)*sqrt(1-4*x)))*g(x*c(x)), c(x) the g.f. of A000108. Hankel transform is A001787(n+1).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000108, A001787, A108081.

Programs

Mathematica

CoefficientList[Series[(1-4*x+(1-x)*Sqrt[1-4*x])/((x+2)*(1-4*x)^(3/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

a(n) = sum(i=0, n, binomial(2*n-i, n+i)*2^i); \\ Michel Marcus, Jul 08 2021

Formula

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

a(n) = Sum_{k=0..n} C(2*n-k,n+k)*2^k.

a(n) = Sum_{k=0..n} C(n+k-1,k)*A001045(n-k+1).

2*n*a(n) +3*(4-5*n)*a(n-1) +6*(4*n-7)*a(n-2) + 8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 14 2011

a(n) ~ 4^n/3. - Vaclav Kotesovec, Oct 20 2012