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%I #17 Jun 22 2022 14:45:24
%S 1,2,15,4140,10480142147,128064670049908713818925644,
%T 172134143357358850934369963665272571125557575184049758045339873395
%N a(n) = A000110(2^n).
%C Number of set partitions of all subsets of a set, Bell(2^n).
%F 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.
%F a(n) = exp(-1) * Sum_{k>=0} k^(2^n)/k!. - _Ilya Gutkovskiy_, Jun 13 2019
%e 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).
%e For S={1,2} we have SU = { {}, {1}, {2}, {1,2} } and W =
%e {
%e {{{}}, {1}, {2}, {1, 2}},
%e {{2}, {1, 2}, {{}, {1}}},
%e {{1}, {1, 2}, {{}, {2}}},
%e {{1}, {2}, {{}, {1, 2}}},
%e {{{}}, {1, 2}, {{1}, {2}}},
%e {{{1}, {2}}, {{}, {1, 2}}},
%e {{1, 2}, {{}, {1}, {2}}},
%e {{{}}, {2}, {{1}, {1, 2}}},
%e {{{1}, {1, 2}}, {{}, {2}}},
%e {{2}, {{}, {1}, {1, 2}}},
%e {{{}}, {1}, {{2}, {1, 2}}},
%e {{{2}, {1, 2}}, {{}, {1}}},
%e {{1}, {{}, {2}, {1, 2}}},
%e {{{}}, {{1}, {2}, {1, 2}}},
%e {{{}, {1}, {2}, {1, 2}}}
%e }
%e and |W| = 15.
%p 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;
%t Table[BellB[2^n],{n,0,10}] (* _Geoffrey Critzer_, Jan 03 2014 *)
%o (Python)
%o from sympy import bell
%o def A135085(n): return bell(2**n) # _Chai Wah Wu_, Jun 22 2022
%Y Cf. A000079, A000110, A008277, A077585, A135084.
%K nonn
%O 0,2
%A _Thomas Wieder_, Nov 18 2007, Nov 19 2007
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