A288787 Number of blocks of size >= five in all set partitions of n.

1, 7, 50, 345, 2392, 16955, 123707, 932010, 7260709, 58509323, 487593202, 4199841037, 37361858716, 342989895895, 3246458915947, 31653980371254, 317654338317380, 3278058775976704, 34757921507150964, 378372365291381716, 4225533329681577846, 48375204740642752562

Offset: 5

Author

Alois P. Heinz, Jun 15 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..575
• Wikipedia, Partition of a set

Crossrefs

Column k=5 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, 5):
seq(a(n), n=5..30);

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, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Formula

a(n) = Bell(n+1) - Sum_{j=0..4} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-5} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..4} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022