A132306 a(n) = Sum_{k=0..2n-1} C(2n-1,k)*trinomial(n,k) for n>0 with a(0)=1.

1, 2, 18, 179, 1874, 20202, 221943, 2470827, 27777618, 314642708, 3585365618, 41054041602, 471980219543, 5444542749674, 62987391100239, 730515277512729, 8490829425196626, 98878672140171984, 1153433769999190212

Offset: 0

Author

Paul D. Hanna, Aug 18 2007

Keywords

• nonn

Comments

Here trinomial(n,k) = A027907(n,k) = [x^k] (1 + x + x^2)^n.

Links

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

Crossrefs

Cf. A082759.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[2n-1,k]*Coefficient[(1+x+x^2)^n,x,k],{k,0,2*n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

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

PARI

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

PARI

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

PARI

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

Formula

a(n) = Sum_{k=0..2n} C(2n,k)*trinomial(n,k)/2 for n>0 with a(0)=1.

a(n) = Sum_{k=0..2n} C(-2n-1,k)*trinomial(n,k)/2 for n>0 with a(0)=1.

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

Recurrence: 2*n*(2*n-1)*a(n) = (39*n^2 - 19*n - 12)*a(n-1) + 2*(53*n^2 - 197*n + 186)*a(n-2) + 12*(n-2)*(2*n-5)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2^(2*n-1)*3^(n+1/2)/sqrt(7*Pi*n) . - Vaclav Kotesovec, Oct 20 2012