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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 Table of n, a(n) for n=0..6. 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 Cf. A000079, A000110, A008277, A077585, A135084. Sequence in context: A337799 A064171 A365628 * 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 May 27 14:41 EDT 2024. Contains 372861 sequences. (Running on oeis4.)