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

%I A135084 #17 Jun 22 2022 14:45:19
%S A135084 1,5,877,1382958545,10293358946226376485095653,
%T A135084 8250771700405624889912456724304738028450190134337110943817172961
%N A135084 a(n) = A000110(2^n-1).
%C A135084 Number of set partitions of all nonempty subsets of a set, Bell(2^n-1).
%H A135084 Amiram Eldar, <a href="/A135084/b135084.txt">Table of n, a(n) for n = 1..9</a>
%F A135084 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.
%e A135084 Let S={1,2,3,...,n} be a set of n elements and let
%e A135084 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).
%e A135084 For S={1,2} we have SU = { {1}, {2}, {1,2} } and W =
%e A135084 {
%e A135084 {{1}, {2}, {1, 2}},
%e A135084 {{1, 2}, {{1}, {2}}},
%e A135084 {{2}, {{1}, {1, 2}}},
%e A135084 {{1}, {{2}, {1, 2}}},
%e A135084 {{{1}, {2}, {1, 2}}}
%e A135084 }
%e A135084 and |W| = 5.
%p A135084 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;
%t A135084 BellB[2^Range[6]-1] (* _Harvey P. Dale_, Jul 22 2012 *)
%o A135084 (Python)
%o A135084 from sympy import bell
%o A135084 def A135084(n): return bell(2**n-1) # _Chai Wah Wu_, Jun 22 2022
%Y A135084 Cf. A000079, A000110, A008277, A077585, A135085.
%K A135084 nonn
%O A135084 1,2
%A A135084 _Thomas Wieder_, Nov 18 2007