A357543 a(n) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2), for n >= 0.

1, 8, 1120, 627200, 896896000, 2611761152000, 13497581633536000, 112839782456360960000, 1427423248072966144000000, 25979103114927983820800000000, 653945983608967208737177600000000, 22056290135163246016287526092800000000, 971138454651237722097139773865984000000000

Offset: 0

Author

Paul D. Hanna, Oct 10 2022

Keywords

• nonn

Comments

Equals row sums of triangle A357540.

a(n) = (3*n+1) * A178575(n) for n >= 0.

Crossrefs

Cf. A357540, A178575, A004117.

Programs

PARI

{a(n) = (3*n+1)!/(3^n*n!) * prod(k=1, n, 3*k-2)}
for(n=0,20, print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} a(n) * x^(3*n+1) / (3*n+1)! = x/(1 - x^3)^(1/3).

a(n) ~ sqrt(2*Pi) * 3^(3*n + 3/2) * n^(3*n + 5/6) / (Gamma(1/3) * exp(3*n)). - Vaclav Kotesovec, Oct 10 2022