A240936 - OEIS

%I #27 Dec 01 2018 21:36:24

%S 1,1,3,21,337,11985,930241,155643329,55638770689,42200814258433,

%T 67536939792143361,227017234854393949185,1596674435594864988020737,

%U 23421099407847007850007154689,714530983411175509576743561314305,45227689798343820164634911814524846081

%N Number of ways to partition the (vertex) set {1,2,...,n} into any number of classes and then select some unordered pairs (edges) <a,b> such that a and b are in distinct classes of the partition.

%C The elements of a class are allowed to be used multiple times to form the unordered pairs.

%C 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.

%C 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

%H Alois P. Heinz, <a href="/A240936/b240936.txt">Table of n, a(n) for n = 0..75</a>

%F 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)).

%F a(n) = Sum_{k=1..n} A058843(n,k) for n>0.

%e 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.

%p b:= proc(n, k) b(n, k):= `if`(k=1, 1, add(binomial(n, i)*

%p       2^(i*(n-i))*b(i, k-1)/k, i=1..n-1))

%p     end:

%p a:= n-> `if`(n=0, 1, add(b(n, k), k=1..n)):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2014

%t 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]

%o (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

%Y Cf. A000569, A001187, A006125, A058843, A277203, A321979.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Aug 03 2014