A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

0, 0, 0, 1, 4, 20, 90, 455, 2352, 13132, 76540, 473660, 3069220, 20922330, 149021600, 1109629885, 8604815520, 69437698160, 581661169640, 5051885815603, 45411759404560, 421977921782270, 4047693372023070, 40034523497947132, 407818256494533984, 4274309903558446900

Offset: 0

Author

Alois P. Heinz, Oct 05 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A327884.

Cf. A000110, A124504, A328155.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0)-b(n, 3):
seq(a(n), n=0..27);

Mathematica

b[n_, k_] := b[n, k] = If[n==0, 1, Sum[If[j==k, 0, b[n-j, k] Binomial[n-1, j-1]], {j, 1, n}]];
a[n_] := b[n, 0] - b[n, 3];
a /@ Range[0, 27] (* Jean-François Alcover, May 02 2020, after Maple *)

Formula

E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^3/6).

a(n) = A000110(n) - A124504(n).