A307903 Coefficient of x^n in (1 + n*x + n*x^3)^n.

1, 1, 4, 36, 448, 6875, 124956, 2624293, 62537728, 1667191653, 49158400000, 1588285928306, 55796298391296, 2117279603738494, 86299754734693696, 3760031421065559375, 174374733095888748544, 8575617145497637681301, 445758339115421869936896, 24417549315693295193935516

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = Sum_{k=0..floor(n/3)} n^(n-2*k) * binomial(n,3*k) * binomial(3*k,k).

a(n) ~ exp(3*n^(1/3)/2^(2/3)) * n^(n - 1/6) / (2^(2/3)*sqrt(3*Pi)) * (1 - 79/(36 * 2^(1/3) * n^(1/3))). - Vaclav Kotesovec, May 05 2019

Mathematica

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

Prog

(PARI) {a(n) = polcoef((1+n*x+n*x^3)^n, n)}
(PARI) {a(n) = sum(k=0, n\3, n^(n-2*k)*binomial(n, 3*k)*binomial(3*k, k))}

Crossrefs

Cf. A092366, A116411, A307904, A307905.

Sequence in context: A167540 A136224 A321963 * A213596 A009533 A135906

Adjacent sequences: A307900 A307901 A307902 * A307904 A307905 A307906

Keyword

nonn

Author

Seiichi Manyama, May 05 2019

Status

approved