A135084 a(n) = A000110(2^n-1).
1, 5, 877, 1382958545, 10293358946226376485095653, 8250771700405624889912456724304738028450190134337110943817172961
Offset: 1
Keywords
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
Programs
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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;
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Mathematica
BellB[2^Range[6]-1] (* Harvey P. Dale, Jul 22 2012 *)
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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.
Comments