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%I #14 Feb 01 2020 22:39:20
%S 1,1,0,1,1,2,3,6,10,21,41,87,187,423,971,2324,5668,14224,36506,95880,
%T 257081,703616,1962887,5578529,16137942,47492141,142093854,432001458,
%U 1333937382,4181500703,13301265585,42918900353,140423545125,465712099790,1565092655597
%N Number of (unlabeled) connected loopless multigraphs such that the sum of the numbers of vertices and edges is n.
%C Also the number of connected skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.
%C a(n) can be computed from A265580 and/or A265581, and partitions of n, by taking all loopless multigraphs (V,E) with |V| + |E| = n and subtracting out the disconnected ones.
%C a(n) <= A265580(n) except when n=1, and a(n) < A265580(n) for n>=6.
%H Andrew Howroyd, <a href="/A265582/b265582.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="https://arxiv.org/abs/1510.06362">Noncrossing partitions, toggles, and homomesies</a>, arXiv:1510.06362 [math.CO], 2015.
%F From _Andrew Howroyd_, Feb 01 2020: (Start)
%F a(n) = Sum_{k=1..ceiling(n/2)} A191646(n-k, k) for n > 0.
%F Inverse Euler transform of A265581. (End)
%e For n = 5, the a(5) = 2 such multigraphs are the graph with three vertices and edges from one vertex to each of the other two, and the graph with two vertices connected by three edges.
%o (PARI) \\ See A191646 for G, InvEulerMT.
%o seq(n)={my(v=InvEulerMT(vector((n+1)\2, k, 1 + y*Ser(G(k, n-1), y)))); Vec(1 + sum(i=1, #v, v[i]*y^i) + O(y*y^n))} \\ _Andrew Howroyd_, Feb 01 2020
%Y Cf. A191646, A265580, A265581.
%K nonn
%O 0,6
%A _Michael Joseph_, Dec 10 2015
%E Terms a(19) and beyond from _Andrew Howroyd_, Feb 01 2020
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