A288786 Number of blocks of size >= four in all set partitions of n.

1, 6, 37, 225, 1395, 8944, 59585, 413117, 2981310, 22380814, 174600298, 1413841252, 11868587577, 103155618776, 927141821215, 8606806236367, 82430269073469, 813600584094320, 8267450613029789, 86406853732930699, 927993270700444588, 10232636504064477996

Offset: 4

Author

Alois P. Heinz, Jun 15 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..575
• Wikipedia, Partition of a set

Crossrefs

Column k=4 of A283424.

Cf. A000110.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
g:= proc(n, k) option remember; `if`(n g(n, 4):
seq(a(n), n=4..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
     `if`(j>3, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=4..30);  # Alois P. Heinz, Jan 06 2022

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k+1] + Binomial[n, k]*b[n - k]];
a[n_] := g[n, 4];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Formula

a(n) = Bell(n+1) - Sum_{j=0..3} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-4} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..3} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 25 2022