cp's OEIS Frontend

A265581 Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.

%I A265581 #18 Feb 01 2020 22:38:32
%S A265581 1,1,1,2,3,5,9,16,29,56,110,222,465,1003,2226,5101,12010,29062,72200,
%T A265581 183886,479544,1279228,3486584,9699975,27520936,79563707,234204235,
%U A265581 701458966,2136296638,6611816700,20784932424,66333327604,214819211047,705650404444,2350231740975
%N A265581 Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.
%C A265581 Also the number of skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.
%C A265581 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 A265581 Andrew Howroyd, <a href="/A265581/b265581.txt">Table of n, a(n) for n = 0..100</a>
%H A265581 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 A265581 a(n) = Sum_{k=0..n} A265580(k).
%F A265581 From _Andrew Howroyd_, Feb 01 2020: (Start)
%F A265581 a(n) = Sum_{i=1..n} A192517(i, n-i) for n > 0.
%F A265581 Euler transform of A265582. (End)
%e A265581 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 A265581 (PARI) \\ Needs G from A191646.
%o A265581 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 A265581 Cf. A191646, A192517, A265580, A265582.
%K A265581 nonn
%O A265581 0,4
%A A265581 _Michael Joseph_, Dec 10 2015
%E A265581 Terms a(19) and beyond from _Andrew Howroyd_, Feb 01 2020