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A265581
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Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.
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3
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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a(n) = Sum_{i=1..n} A192517(i, n-i) for n > 0.
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EXAMPLE
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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.
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PROG
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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