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%I #18 Feb 01 2020 22:38:32
%S 1,1,1,2,3,5,9,16,29,56,110,222,465,1003,2226,5101,12010,29062,72200,
%T 183886,479544,1279228,3486584,9699975,27520936,79563707,234204235,
%U 701458966,2136296638,6611816700,20784932424,66333327604,214819211047,705650404444,2350231740975
%N Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.
%C Also the number of skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.
%C a(n) is the sum of A265580(k) as k ranges from 0 to n. This is because there is a bijection between loopless multigraphs (V,E) satisfying |V| + |E| = k with no isolated vertices and loopless multigraphs (V,E) satisfying |V| + |E| = n with exactly n-k isolated vertices.
%H Andrew Howroyd, <a href="/A265581/b265581.txt">Table of n, a(n) for n = 0..100</a>
%H D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp and S. Rubinstein-Salzedo, <a href="http://arxiv.org/abs/1510.06362">Noncrossing partitions, toggles, and homomesies</a>, arXiv:1510.06362 [math.CO], 2015.
%F a(n) = Sum_{k=0..n} A265580(k).
%F From _Andrew Howroyd_, Feb 01 2020: (Start)
%F a(n) = Sum_{i=1..n} A192517(i, n-i) for n > 0.
%F Euler transform of A265582. (End)
%e For n = 4, the a(4) = 3 such multigraphs are the graph with four isolated vertices, the graph with three vertices and an edge between two of them, and the graph with two vertices connected by two edges.
%o (PARI) \\ Needs G from A191646.
%o seq(n)={vector(n+1,i,1) + sum(k=1, n, concat(vector(n-k+1), G(n-k, k)))} \\ _Andrew Howroyd_, Feb 01 2020
%Y Cf. A191646, A192517, A265580, A265582.
%K nonn
%O 0,4
%A _Michael Joseph_, Dec 10 2015
%E Terms a(19) and beyond from _Andrew Howroyd_, Feb 01 2020
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