A229333 Product of sizes of all the nonempty subsets of an n-element set.

1, 1, 2, 24, 20736, 309586821120, 11501279977342425366528000000, 115744510977565557983391999957434605749927936000000000000000000000

Offset: 0

Author

Jaroslav Krizek, Sep 29 2013

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10
• Aaron R. Bagheri, Classifying the Jacobian Groups of Adinkras, (2017), HMC Senior Theses.

Crossrefs

Cf. A001787 (total size of all the subsets of an n-element set).

Programs

Maple

a:= n-> mul(k^binomial(n, k), k=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Oct 05 2014

Mathematica

Table[Times @@ Rest[Length /@ Subsets[Range[n]]], {n, 7}] (* T. D. Noe, Oct 01 2013 *)

Python

from math import comb, prod
def a(n): return prod(k**comb(n, k) for k in range(1, n+1))
print([a(n) for n in range(11)]) # Michael S. Branicky, Jun 25 2022

Formula

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