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A135085 a(n) = A000110(2^n). 4
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
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
Sequence in context: A337799 A064171 A365628 * A290042 A080911 A175981
KEYWORD
nonn
AUTHOR
Thomas Wieder, Nov 18 2007, Nov 19 2007
STATUS
approved

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Last modified May 27 14:41 EDT 2024. Contains 372861 sequences. (Running on oeis4.)