cp's OEIS Frontend

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

%I A143140 #32 Mar 19 2018 04:06:53
%S A143140 1,1,2,11,83,787,8965,119170,1810450,30942699,587606593,12274606775,
%T A143140 279715819531,6905395692990,183588212652382,5229549060414223,
%U A143140 158895798308201987,5129671140284343035,175343720698891809337,6326623756471457351814,240286954202031694593966
%N A143140 Total number of all repeated partitions of the n-set {1,2,3,...,n}.
%C A143140 The corresponding unlabeled counterpart is sequence A143141.
%C A143140 See also A131407 = Repeated set partitions or nested set partitions. Possible coalitions among n persons.
%C A143140 See also A137731 = Repeated set splitting, labeled elements.
%C A143140 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
%H A143140 Alois P. Heinz, <a href="/A143140/b143140.txt">Table of n, a(n) for n = 0..407</a>
%F A143140 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
%e A143140 a(1) = |{{{1}}}| = 1.
%e A143140 a(2) = |{{{1,2}}, {{1},{2}}}| = 2.
%e A143140 a(3) = |{{{1,2,3}}, {{1,2},{3}}, {{2},{1,3}}, {{1},{2,3}}, {{1},{2},{3}}}| + 3*a(2) = 5 + 3*2 = 11.
%p A143140 with(combinat):
%p A143140 a:= proc(n) option remember;
%p A143140       bell(n)+ add(a(k)*binomial(n, k)*bell(n-k), k=2..n-1)
%p A143140     end:
%p A143140 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 26 2012
%t A143140 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_ *)
%Y A143140 Cf. A000110, A131407, A137731, A143141.
%K A143140 nonn
%O A143140 0,3
%A A143140 _Thomas Wieder_, Jul 27 2008
%E A143140 Edited by _Thomas Wieder_, Jul 26 2012
%E A143140 More terms from _Alois P. Heinz_, Jul 26 2012
%E A143140 a(0)=1 prepended by _Alois P. Heinz_, Sep 22 2016