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 A265581 Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n. 3
 1, 1, 1, 2, 3, 5, 9, 16, 29, 56, 110, 222, 465, 1003, 2226, 5101, 12010, 29062, 72200, 183886, 479544, 1279228, 3486584, 9699975, 27520936, 79563707, 234204235, 701458966, 2136296638, 6611816700, 20784932424, 66333327604, 214819211047, 705650404444, 2350231740975 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also the number of skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below. 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..100 D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp and S. Rubinstein-Salzedo, Noncrossing partitions, toggles, and homomesies, arXiv:1510.06362 [math.CO], 2015. FORMULA a(n) = Sum_{k=0..n} A265580(k). From Andrew Howroyd, Feb 01 2020: (Start) a(n) = Sum_{i=1..n} A192517(i, n-i) for n > 0. Euler transform of A265582. (End) EXAMPLE 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. PROG (PARI) \\ Needs G from A191646. 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 CROSSREFS Cf. A191646, A192517, A265580, A265582. Sequence in context: A198518 A182558 A298204 * A335703 A107250 A050168 Adjacent sequences:  A265578 A265579 A265580 * A265582 A265583 A265584 KEYWORD nonn AUTHOR Michael Joseph, Dec 10 2015 EXTENSIONS Terms a(19) and beyond from Andrew Howroyd, Feb 01 2020 STATUS approved

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Last modified July 2 10:39 EDT 2022. Contains 355004 sequences. (Running on oeis4.)