

A229333


Product of sizes of all the nonempty subsets of an nelement set.


2




OFFSET

0,3


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


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.


FORMULA

a(n) = Product_{k=1..n} k^C(n,k) = Product_{k=1..n} k^(n!/((nk)!*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


EXAMPLE

For n=3; nonempty subsets of 3element 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!/((31)!*1!))] * [2^(3!/((32)!*2!))] * [3^(3!/((33)!*3!))] = 1^3 * 2^3 * 3^1 = 24.


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 *)


CROSSREFS

Cf. A001787 (total size of all the subsets of an nelement set).
Sequence in context: A152687 A062716 A137888 * A108349 A000722 A098679
Adjacent sequences: A229330 A229331 A229332 * A229334 A229335 A229336


KEYWORD

nonn


AUTHOR

Jaroslav Krizek, Sep 29 2013


STATUS

approved



