A168595 a(n) = Sum_{k=0..2n} C(2n,k)*A027907(n,k) where A027907 is the triangle of trinomial coefficients.

1, 4, 36, 358, 3748, 40404, 443886, 4941654, 55555236, 629285416, 7170731236, 82108083204, 943960439086, 10889085499348, 125974782200478, 1461030555025458, 16981658850393252, 197757344280343968

Offset: 0

Author

Paul D. Hanna, Nov 30 2009

Keywords

• nonn

Comments

Compare to A092765(n) = Sum_{k=0..2n} (-1)^k*C(2n,k)*A027907(n,k), which is the number of paths of length n ending at origin in 1-D random walk with jumps to next-nearest neighbors.

Crossrefs

Cf. A027907, A092765.

Programs

Maple

cb := n -> binomial(2*n, n);
a := n -> add((-1)^(n-k)*binomial(n,k)*cb(n+k), k=0..n);
seq(a(n), n=0..17); # Peter Luschny, Aug 15 2017

PARI

{a(n)=sum(k=0,2*n,binomial(2*n,k)*polcoeff((1+x+x^2)^n,k))}

Formula

a(n) = 2*A132306(n) for n > 0. - Mark van Hoeij, Jul 02 2010

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cb(n+k) with cb(n) = binomial(2n,n). - Peter Luschny, Aug 15 2017