A094762 a(n) = Bell(n+1) - 2^n + 1 + n, where Bell(i) is the i-th Bell number A000110(i).

1, 2, 4, 11, 41, 177, 820, 4020, 20900, 115473, 677557, 4211561, 27640354, 190891144, 1382942176, 10480109395, 82864804285, 682076675105, 5832741942932, 51724157711104, 474869815108196, 4506715736350193, 44152005850890065, 445958869286416705

Offset

0,2

Comments

a(n) is the solution to the following combinatorial problem. Given a set S of n labeled elements, find the number of all subsets of S (2^n) plus the number of partitions of any subset T of S into parts which are not all of size 1 nor of size |T|. This implies that a(n) = 2^n + Sum_{m=3..n} (Bell(m)-2) = Bell(n+1) - 2^n + 1 + n, using the standard recurrence for the Bell numbers (Comtet, Eq. (4a)). - N. J. A. Sloane, Nov 26 2013

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 210.

Links

Table of n, a(n) for n=0..23.

Formula

Also a(n) = 2^n + Sum_{m=3..n} binomial(n,m)*(Bell(m)-2).

Maple

with(combinat); [seq(bell(n+1)-2^n+n+1, n=0..30)];

Mathematica

Table[BellB[n+1]-2^n+n+1, {n, 0, 30}] (* Harvey P. Dale, Apr 24 2018 *)

Crossrefs

Cf. A000110, A144646.

Sequence in context: A012948 A013182 A013170 * A295772 A099934 A067353

Adjacent sequences: A094759 A094760 A094761 * A094763 A094764 A094765

Keyword

nonn

Author

Jean-Yves Tallet and N. J. A. Sloane, Jan 24 2009

Status

approved