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

0,4

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