OFFSET
0,3
COMMENTS
The elements of a class are allowed to be used multiple times to form the unordered pairs.
Equivalently, a(n) is the sum of the number of k-colored graphs on n labeled nodes taken over k colors, 1<=k<=n, where labeled graphs using k colors that differ only by a permutation of the k colors are considered to be the same.
Also the number of ways to choose a stable partition of a simple graph on n vertices. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. - Gus Wiseman, Nov 24 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..75
FORMULA
a(n) = n! * 2^C(n,2) * [x^n] exp(E(x)-1) where E(x) is Sum_{n>=0} x^n/(n!*2^C(n,2)).
a(n) = Sum_{k=1..n} A058843(n,k) for n>0.
EXAMPLE
a(2)=3 because the empty graph with 2 nodes is counted twice (once for each partition of 2) and the complete graph is counted once. 2+1=3.
MAPLE
b:= proc(n, k) b(n, k):= `if`(k=1, 1, add(binomial(n, i)*
2^(i*(n-i))*b(i, k-1)/k, i=1..n-1))
end:
a:= n-> `if`(n=0, 1, add(b(n, k), k=1..n)):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 04 2014
MATHEMATICA
nn=15; e[x_]:=Sum[x^n/(n!*2^Binomial[n, 2]), {n, 0, nn}]; Table[n!2^Binomial[n, 2], {n, 0, nn}]CoefficientList[Series[Exp[(e[x]-1)], {x, 0, nn}], x]
PROG
(PARI) seq(n)={Vec(serconvol(sum(j=0, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n), exp(sum(j=1, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))))} \\ Andrew Howroyd, Dec 01 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Aug 03 2014
STATUS
approved