A327504 Number of set partitions of [n] where each subset is again partitioned into three nonempty subsets.

1, 0, 0, 1, 6, 25, 100, 511, 3626, 29765, 250200, 2146771, 19575446, 195336505, 2124840900, 24646324431, 299803782466, 3809251939245, 50698296967600, 708349718638891, 10372758309704686, 158546862369781985, 2519789706502636700, 41545703617137280551

Offset: 0

Author

Alois P. Heinz, Sep 14 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..501
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A324162.

Cf. A346894.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 3), j=3..n))
    end:
seq(a(n), n=0..25);

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j -1] StirlingS2[j, 3], {j, 3, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

PARI

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/(6^k*k!)); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f.: exp((exp(x)-1)^3/3!).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(6^k * k!). - Seiichi Manyama, May 07 2022