A307905 Coefficient of x^n in (1 + n*x + x^3)^n.
1, 1, 4, 30, 304, 3875, 59631, 1076383, 22309120, 522262245, 13631508400, 392535959156, 12362973152751, 422774554883590, 15600699362473876, 617888566413340503, 26145122799198386944, 1177107512023013681429, 56185125998674634494980, 2834081165961033246374350
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..385
Programs
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Maple
f:= n -> coeff((1+n*x+x^3)^n,x,n): map(f, [$0..30]); # Robert Israel, Mar 27 2023
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Mathematica
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 *) -
PARI
{a(n) = polcoef((1+n*x+x^3)^n, n)} -
PARI
{a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n,3*k)*binomial(3*k,k))}
Formula
a(n) = Sum_{k=0..floor(n/3)} n^(n-3*k) * binomial(n,3*k) * binomial(3*k,k).
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