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A135085 a(n) = A000110(2^n). 3
1, 2, 15, 4140, 10480142147, 128064670049908713818925644, 172134143357358850934369963665272571125557575184049758045339873395 (list; graph; refs; listen; history; text; internal format)
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((stirling2(2^n,k)), k=0..2^n) = 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 *)

CROSSREFS

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

Sequence in context: A320768 A337799 A064171 * A290042 A080911 A175981

Adjacent sequences:  A135082 A135083 A135084 * A135086 A135087 A135088

KEYWORD

nonn

AUTHOR

Thomas Wieder, Nov 18 2007, Nov 19 2007

STATUS

approved

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Last modified September 17 03:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)