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A003465
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Number of ways to cover an n-set.
(Formerly M4024)
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137
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1, 1, 5, 109, 32297, 2147321017, 9223372023970362989, 170141183460469231667123699502996689125, 57896044618658097711785492504343953925273862865136528166133547991141168899281
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OFFSET
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0,3
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COMMENTS
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Let S be an n-element set, and let P be the set of all nonempty subsets of S. Then a(n) = number of subsets of P whose union is S.
Including the empty set doubles the entries, and we get A000371.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. G. Wagner, Covers of finite sets, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 515-520.
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LINKS
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D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Eric Weisstein's World of Mathematics, Cover.
Yoad Winter and Remko Scha, Plurals, draft chapter for the Wiley-Blackwell Handbook of Contemporary Semantics - second edition, edited by Shalom Lappin and Chris Fox, 2014.
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FORMULA
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a(n) = Sum_{k>=0} (-1)^k * binomial(n, k) * 2^(2^(n-k)) / 2. - Michael Somos, Jun 14 1999
E.g.f.: (1/2)*Sum_{n>=0} exp((2^n-1)*x)*log(2)^n/n!. - Vladeta Jovovic, May 30 2004
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EXAMPLE
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Let n=2, S={a,b}, P={a,b,ab}. There are five subsets of P whose union is S: {ab}, {a,b}, {a,ab}, {b,ab}, {a,b,ab}. - Marc LeBrun, Nov 10 2010
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MAPLE
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a:= n-> add((-1)^k * binomial(n, k)*2^(2^(n-k))/2, k=0..n):
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MATHEMATICA
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Table[Sum[(-1)^j Binomial[n, j] 2^(2^(n-j)-1), {j, 0, n}], {n, 0, 10}] (* Geoffrey Critzer, Jun 26 2013 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, (-1)^k * n!/k!/(n-k)! * 2^(2^(n-k))) / 2} /* Michael Somos, Jun 14 1999 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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