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A143140
Total number of all repeated partitions of the n-set {1,2,3,...,n}.
2
1, 1, 2, 11, 83, 787, 8965, 119170, 1810450, 30942699, 587606593, 12274606775, 279715819531, 6905395692990, 183588212652382, 5229549060414223, 158895798308201987, 5129671140284343035, 175343720698891809337, 6326623756471457351814, 240286954202031694593966
OFFSET
0,3
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
LINKS
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Jul 27 2008
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
STATUS
approved