A383881 a(n) = [x^n] Product_{k=1..3*n} 1/(1 - k*x).

1, 6, 266, 22275, 2757118, 452329200, 92484925445, 22653141490980, 6466506598695390, 2108114165258886708, 772778072287000494520, 314641228029527540596455, 140880584836935832288402135, 68799366730032076856334789900, 36392216443342587869022660451080, 20728132932716479897744043460870000

Offset: 0

Author

Vaclav Kotesovec, May 13 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A007820, A348084, A383882.

Cf. A217913.

Programs

Magma

[&+[Abs(StirlingSecond(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

Mathematica

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 3*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[4*n, 3*n], {n, 0, 15}]
Table[SeriesCoefficient[(-1)^n/(Pochhammer[1 - 1/x, 3*n]*x^(3*n)), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = Stirling2(4*n,3*n).

a(n) ~ (-1)^(3*n) * 4^(4*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 3^(3*n + 1/2) * w^(3*n) * (4/3 + w)^n), where w = LambertW(-4/(3*exp(4/3))).