OFFSET
0,2
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
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
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 *)
PROG
(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))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 18 2007
STATUS
approved