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

1, 5, 877, 1382958545, 10293358946226376485095653, 8250771700405624889912456724304738028450190134337110943817172961

Offset

1,2

Comments

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

Links

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

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.

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A332185 A085706 A190350 * A206356 A298278 A299371

Adjacent sequences: A135081 A135082 A135083 * A135085 A135086 A135087

Keyword

nonn

Author

Thomas Wieder, Nov 18 2007

Status

approved