A265582 Number of (unlabeled) connected loopless multigraphs such that the sum of the numbers of vertices and edges is n.

1, 1, 0, 1, 1, 2, 3, 6, 10, 21, 41, 87, 187, 423, 971, 2324, 5668, 14224, 36506, 95880, 257081, 703616, 1962887, 5578529, 16137942, 47492141, 142093854, 432001458, 1333937382, 4181500703, 13301265585, 42918900353, 140423545125, 465712099790, 1565092655597

Offset: 0

Author

Michael Joseph, Dec 10 2015

Keywords

• nonn

Comments

Also the number of connected skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.

a(n) can be computed from A265580 and/or A265581, and partitions of n, by taking all loopless multigraphs (V,E) with |V| + |E| = n and subtracting out the disconnected ones.

a(n) <= A265580(n) except when n=1, and a(n) < A265580(n) for n>=6.

Examples

			For n = 5, the a(5) = 2 such multigraphs are the graph with three vertices and edges from one vertex to each of the other two, and the graph with two vertices connected by three 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, A265580, A265581.

Programs

PARI

\\ See A191646 for G, InvEulerMT.
seq(n)={my(v=InvEulerMT(vector((n+1)\2, k, 1 + y*Ser(G(k, n-1), y)))); Vec(1 + sum(i=1, #v, v[i]*y^i) + O(y*y^n))} \\ Andrew Howroyd, Feb 01 2020

Formula

From Andrew Howroyd, Feb 01 2020: (Start)

a(n) = Sum_{k=1..ceiling(n/2)} A191646(n-k, k) for n > 0.

Inverse Euler transform of A265581. (End)

Extensions

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