A288788 Number of blocks of size >= 6 in all set partitions of n.

1, 8, 65, 502, 3851, 29921, 237426, 1932529, 16173029, 139320277, 1235847277, 11288120480, 106132359679, 1026681599731, 10212591089574, 104393925768077, 1095895294558168, 11806719056706773, 130457490607638988, 1477428802636263486, 17138268233851671782

Offset

6,2

Links

Alois P. Heinz, Table of n, a(n) for n = 6..575

Wikipedia, Partition of a set

Formula

a(n) = Bell(n+1) - Sum_{j=0..5} binomial(n,j) * Bell(n-j).

a(n) = Sum_{j=0..n-6} binomial(n,j) * Bell(j).

E.g.f.: (exp(x) - Sum_{k=0..5} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022

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<k, 0,
      g(n, k+1) +binomial(n, k)*b(n-k))
    end:
a:= n-> g(n, 6):
seq(a(n), n=6..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, 6];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

Crossrefs

Column k=6 of A283424.

Cf. A000110.

Sequence in context: A293802 A316872 A317600 * A033118 A033126 A022039

Adjacent sequences: A288785 A288786 A288787 * A288789 A288790 A288791

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2017

Status

approved