cp's OEIS Frontend

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

%I A132306 #10 May 25 2013 07:23:01
%S A132306 1,2,18,179,1874,20202,221943,2470827,27777618,314642708,3585365618,
%T A132306 41054041602,471980219543,5444542749674,62987391100239,
%U A132306 730515277512729,8490829425196626,98878672140171984,1153433769999190212
%N A132306 a(n) = Sum_{k=0..2n-1} C(2n-1,k)*trinomial(n,k) for n>0 with a(0)=1.
%C A132306 Here trinomial(n,k) = A027907(n,k) = [x^k] (1 + x + x^2)^n.
%H A132306 Vincenzo Librandi, <a href="/A132306/b132306.txt">Table of n, a(n) for n = 0..200</a>
%F A132306 a(n) = Sum_{k=0..2n} C(2n,k)*trinomial(n,k)/2 for n>0 with a(0)=1.
%F A132306 a(n) = Sum_{k=0..2n} C(-2n-1,k)*trinomial(n,k)/2 for n>0 with a(0)=1.
%F A132306 a(n) = Sum_{k=0..2n} C(-2n,k)*trinomial(n,k).
%F A132306 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 A132306 a(n) ~ 2^(2*n-1)*3^(n+1/2)/sqrt(7*Pi*n) . - _Vaclav Kotesovec_, Oct 20 2012
%t A132306 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 A132306 (PARI) {a(n)=sum(k=0,2*n,binomial(2*n-1,k)*polcoeff((1+x+x^2)^n,k))}
%o A132306 (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 A132306 (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 A132306 (PARI) {a(n)=sum(k=0,2*n,binomial(-2*n,k)*polcoeff((1+x+x^2)^n,k))}
%Y A132306 Cf. A082759.
%K A132306 nonn
%O A132306 0,2
%A A132306 _Paul D. Hanna_, Aug 18 2007