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

0,3

Comments

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

Links

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

Formula

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

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

Example

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}}

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}]

Prog

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

Crossrefs

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

Sequence in context: A243951 A290941 A211051 * A191962 A326650 A323572

Adjacent sequences: A322658 A322659 A322660 * A322662 A322663 A322664

Keyword

nonn

Author

Gus Wiseman, Dec 22 2018

Status

approved