A322661 Number of graphs with loops spanning n labeled vertices.

1, 1, 5, 45, 809, 28217, 1914733, 254409765, 66628946641, 34575388318705, 35680013894626133, 73392583417010454429, 301348381381966079690489, 2471956814761996896091805993, 40530184362443281653842556898237, 1328619783326799871943604598592805525

Offset: 0

Author

Gus Wiseman, Dec 22 2018

Keywords

• nonn

Comments

The span of a graph is the union of its edges.

Examples

			The a(2) = 5 edge-sets:
  {{1,2}}
  {{1,1},{1,2}}
  {{1,1},{2,2}}
  {{1,2},{2,2}}
  {{1,1},{1,2},{2,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000666, A006125, A006129 (loops not allowed), A054921, A062740, A116539, A320461, A322635, A048291 (for directed edgs).

Programs

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[n,k]*2^Binomial[k+1,2],{k,0,n}],{n,10}]
(* second program *)
Table[Select[Expand[Product[1+x[i]*x[j],{j,n},{i,j}]],And@@Table[!FreeQ[#,x[i]],{i,n}]&]/.x[_]->1,{n,7}]

PARI

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*2^binomial(k+1,2)) \\ Andrew Howroyd, Jan 06 2024

Formula

Exponential transform of A062740, if we assume A062740(1) = 1.

Inverse binomial transform of A006125(n+1) = 2^binomial(n+1,2).