A307905 - OEIS

%I #19 Mar 28 2023 08:00:03

%S 1,1,4,30,304,3875,59631,1076383,22309120,522262245,13631508400,

%T 392535959156,12362973152751,422774554883590,15600699362473876,

%U 617888566413340503,26145122799198386944,1177107512023013681429,56185125998674634494980,2834081165961033246374350

%N Coefficient of x^n in (1 + n*x + x^3)^n.

%H Robert Israel, <a href="/A307905/b307905.txt">Table of n, a(n) for n = 0..385</a>

%F a(n) = Sum_{k=0..floor(n/3)} n^(n-3*k) * binomial(n,3*k) * binomial(3*k,k).

%F a(n) ~ c * n^n, where c = Sum_{k>=0} 1/(k!*(2*k)!) = HypergeometricPFQ[{}, {1/2, 1}, 1/4] = 1.52106585051363080966025715155941607334728986626976774617... - _Vaclav Kotesovec_, May 05 2019

%p f:= n -> coeff((1+n*x+x^3)^n,x,n):

%p map(f, [$0..30]); # _Robert Israel_, Mar 27 2023

%t Flatten[{1, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27/(4*n^3)], {n, 1, 20}]}] (* _Vaclav Kotesovec_, May 05 2019 *)

%o (PARI) {a(n) = polcoef((1+n*x+x^3)^n, n)}

%o (PARI) {a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n,3*k)*binomial(3*k,k))}

%Y Cf. A116411, A186925, A307903, A307904.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 05 2019