A132306 - OEIS

%I #10 May 25 2013 07:23:01

%S 1,2,18,179,1874,20202,221943,2470827,27777618,314642708,3585365618,

%T 41054041602,471980219543,5444542749674,62987391100239,

%U 730515277512729,8490829425196626,98878672140171984,1153433769999190212

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

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

%H Vincenzo Librandi, <a href="/A132306/b132306.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

%F 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

%F a(n) ~ 2^(2*n-1)*3^(n+1/2)/sqrt(7*Pi*n) . - _Vaclav Kotesovec_, Oct 20 2012

%t 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 *)

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

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

%o (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)}

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

%Y Cf. A082759.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 18 2007