A135084 a(n) = A000110(2^n-1).

1, 5, 877, 1382958545, 10293358946226376485095653, 8250771700405624889912456724304738028450190134337110943817172961

Offset: 1

Author

Thomas Wieder, Nov 18 2007

Keywords

• nonn

Comments

Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).

Examples

			Let S={1,2,3,...,n} be a set of n elements and let
SU be the set of all nonempty subsets of S. The number of elements of SU is |SU| = 2^n-1. Now form all possible set partitions from SU where the empty set is excluded. This gives a set W and its number of elements is |W| = Sum_{k=1..2^n-1} Stirling2(2^n-1,k).
For S={1,2} we have SU = { {1}, {2}, {1,2} } and W =
{
{{1}, {2}, {1, 2}},
{{1, 2}, {{1}, {2}}},
{{2}, {{1}, {1, 2}}},
{{1}, {{2}, {1, 2}}},
{{{1}, {2}, {1, 2}}}
}
and |W| = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..9

Crossrefs

Cf. A000079, A000110, A008277, A077585, A135085.

Programs

Maple

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n,nend,arg,k,w; nend:=5; for n from 1 to nend do arg:=2^n-1; w[n]:=sum((stirling2(arg,k)), k=1..arg); od; print(w[1],w[2],w[3],w[4],w[5],w[6],w[7],w[8],w[9],w[10]); end proc;

Mathematica

BellB[2^Range[6]-1] (* Harvey P. Dale, Jul 22 2012 *)

Python

from sympy import bell
def A135084(n): return bell(2**n-1) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = Sum_{k=1..2^n-1} Stirling2(2^n-1,k) = Bell(2^n-1), where Stirling2(n, k) is the Stirling number of the second kind and Bell(n) is the Bell number.