A168595 - OEIS

%I #7 Aug 15 2017 13:41:41

%S 1,4,36,358,3748,40404,443886,4941654,55555236,629285416,7170731236,

%T 82108083204,943960439086,10889085499348,125974782200478,

%U 1461030555025458,16981658850393252,197757344280343968

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

%C 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.

%F a(n) = 2*A132306(n) for n > 0. - _Mark van Hoeij_, Jul 02 2010

%F 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

%p cb := n -> binomial(2*n, n);

%p a := n -> add((-1)^(n-k)*binomial(n,k)*cb(n+k), k=0..n);

%p seq(a(n), n=0..17); # _Peter Luschny_, Aug 15 2017

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

%Y Cf. A027907, A092765.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 30 2009