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

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

Offset: 0

Author

Michael Joseph, Dec 10 2015

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A191646, A192517, A265580, A265582.

Programs

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

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)

Extensions

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2020