A307903 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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)}