A135085 a(n) = A000110(2^n).

1, 2, 15, 4140, 10480142147, 128064670049908713818925644, 172134143357358850934369963665272571125557575184049758045339873395

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..6.

Formula

a(n) = |W| = Sum_{k=0..2^n} Stirling2(2^n,k) = Bell(2^n), where Stirling2(n) is the Stirling number of the second kind and Bell(n) is the Bell number.

a(n) = exp(-1) * Sum_{k>=0} k^(2^n)/k!. - Ilya Gutkovskiy, Jun 13 2019

Example

Let S={1,2,3,...,n} be a set of n elements and let SU be the set of all subsets of S including the empty set. The number of elements of SU is |SU| = 2^n. Now form all possible set partitions from SU including the empty set. This gives a set W and its number of elements is |W| = sum((stirling2(2^n,k)), k=0..2^n) = Bell(2^n).
For S={1,2} we have SU = { {}, {1}, {2}, {1,2} } and W =
{
{{{}}, {1}, {2}, {1, 2}},
{{2}, {1, 2}, {{}, {1}}},
{{1}, {1, 2}, {{}, {2}}},
{{1}, {2}, {{}, {1, 2}}},
{{{}}, {1, 2}, {{1}, {2}}},
{{{1}, {2}}, {{}, {1, 2}}},
{{1, 2}, {{}, {1}, {2}}},
{{{}}, {2}, {{1}, {1, 2}}},
{{{1}, {1, 2}}, {{}, {2}}},
{{2}, {{}, {1}, {1, 2}}},
{{{}}, {1}, {{2}, {1, 2}}},
{{{2}, {1, 2}}, {{}, {1}}},
{{1}, {{}, {2}, {1, 2}}},
{{{}}, {{1}, {2}, {1, 2}}},
{{{}, {1}, {2}, {1, 2}}}
}
and |W| = 15.

Maple

ZahlDerMengenAusMengeDerZerlegungenEinerMenge:=proc() local n, nend, arg, k, w; nend:=5; for n from 0 to nend do arg:=2^n; w[n]:=sum((stirling2(arg, k)), k=0..arg); od; print(w[0], w[1], w[2], w[3], w[4], w[5], w[6], w[7], w[8], w[9], w[10]); end proc;

Mathematica

Table[BellB[2^n], {n, 0, 10}] (* Geoffrey Critzer, Jan 03 2014 *)

Prog

(Python)
from sympy import bell
def A135085(n): return bell(2**n) # Chai Wah Wu, Jun 22 2022

Crossrefs

Cf. A000079, A000110, A008277, A077585, A135084.

Sequence in context: A337799 A064171 A365628 * A290042 A080911 A175981

Adjacent sequences: A135082 A135083 A135084 * A135086 A135087 A135088

Keyword

nonn

Author

Thomas Wieder, Nov 18 2007, Nov 19 2007

Status

approved