A143140 Total number of all repeated partitions of the n-set {1,2,3,...,n}.

1, 1, 2, 11, 83, 787, 8965, 119170, 1810450, 30942699, 587606593, 12274606775, 279715819531, 6905395692990, 183588212652382, 5229549060414223, 158895798308201987, 5129671140284343035, 175343720698891809337, 6326623756471457351814, 240286954202031694593966

Offset: 0

Author

Thomas Wieder, Jul 27 2008

Keywords

• nonn

Comments

The corresponding unlabeled counterpart is sequence A143141.

See also A131407 = Repeated set partitions or nested set partitions. Possible coalitions among n persons.

See also A137731 = Repeated set splitting, labeled elements.

a(n) is the number of set partitions of the n-set plus sum of a(k) for all the k-sets (1 < k < n) that are contained (with multiplicity) in these set partitions. - Alois P. Heinz, Jul 27 2012

Examples

			a(1) = |{{{1}}}| = 1.
a(2) = |{{{1,2}}, {{1},{2}}}| = 2.
a(3) = |{{{1,2,3}}, {{1,2},{3}}, {{2},{1,3}}, {{1},{2,3}}, {{1},{2},{3}}}| + 3*a(2) = 5 + 3*2 = 11.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..407

Crossrefs

Cf. A000110, A131407, A137731, A143141.

Programs

Maple

with(combinat):
a:= proc(n) option remember;
      bell(n)+ add(a(k)*binomial(n, k)*bell(n-k), k=2..n-1)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 26 2012

Mathematica

a[n_] := a[n] = BellB[n]+Sum[a[k]*Binomial[n, k]*BellB[n-k], {k, 2, n-1}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

Formula

a(n) = Bell(n) + Sum_{k=2..n-1} C(n,k)*Bell(n-k)*a(k) with Bell = A000110. - Alois P. Heinz, Jul 26 2012

Extensions

Edited by Thomas Wieder, Jul 26 2012

More terms from Alois P. Heinz, Jul 26 2012

a(0)=1 prepended by Alois P. Heinz, Sep 22 2016