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A229333
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Product of sizes of all the nonempty subsets of an n-element set.
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4
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OFFSET
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0,3
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COMMENTS
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Equivalently, a(n) is the number of functions from the nonempty subsets of {1,2,...,n} into {1,2,...,n} such that each subset is mapped to an element that it contains. - Geoffrey Critzer, Oct 05 2014
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} k^C(n,k) = Product_{k=1..n} k^(n!/((n-k)!*k!)).
log(a(n)) ~ 2^n*(log(n/2) - 1/(2*n) - 3/(4*n^2) - 2/n^3 - 65/(8*n^4) - 134/(3*n^5) - 1239/(4*n^6) - 2594/n^7 - 407409/(16*n^8) - 1433418/(5*n^9) - 14565881/(4*n^10) - ...). - Vaclav Kotesovec, Jul 24 2015
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EXAMPLE
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For n=3; nonempty subsets of a 3-element set: {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}; product of numbers of elements of these subsets = 1*1*1*2*2*2*3 = 24. For n = 3; a(3) = [1^(3!/((3-1)!*1!))] * [2^(3!/((3-2)!*2!))] * [3^(3!/((3-3)!*3!))] = 1^3 * 2^3 * 3^1 = 24.
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MAPLE
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a:= n-> mul(k^binomial(n, k), k=1..n):
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MATHEMATICA
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Table[Times @@ Rest[Length /@ Subsets[Range[n]]], {n, 7}] (* T. D. Noe, Oct 01 2013 *)
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PROG
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(Python)
from math import comb, prod
def a(n): return prod(k**comb(n, k) for k in range(1, n+1))
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CROSSREFS
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Cf. A001787 (total size of all the subsets of an n-element set).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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