A307904 Coefficient of x^n in (1 + x + n*x^3)^n.

1, 1, 1, 10, 49, 151, 901, 5881, 28225, 165565, 1158601, 6993196, 44201521, 320103070, 2200745821, 15118248601, 113390231809, 845797019077, 6250243032145, 48718551529210, 384815404148401, 3021055319338813, 24492293678972725, 202303201125303565, 1669594463059152961

Offset

0,4

Links

Table of n, a(n) for n=0..24.

Formula

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

log(a(n)) ~ (n/3 - 1/2)*log(n) + (log(3) - 2*log(2)/3)*n + (2*n)^(2/3)/3 - (2*n)^(1/3)/9. - Vaclav Kotesovec, May 05 2019

Mathematica

Table[HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27*n/4], {n, 0, 25}] (* Vaclav Kotesovec, May 05 2019 *)

Prog

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

Crossrefs

Cf. A116411, A187018, A307903, A307905.

Sequence in context: A154066 A056578 A370216 * A226797 A163716 A271662

Adjacent sequences: A307901 A307902 A307903 * A307905 A307906 A307907

Keyword

nonn

Author

Seiichi Manyama, May 05 2019

Status

approved