A322698 Number of regular graphs with half-edges on n labeled vertices.

1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610

Offset

0,2

Comments

A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.

Links

Table of n, a(n) for n=0..18.

Wikipedia, Regular graph

Example

The a(3) = 10 edge sets:
  {}
  {{1},{2,3}}
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Mathematica

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s, {s, Union/@Select[Tuples[Range[n], 2], OrderedQ]}], Sequence@@Table[{x[i], 0, k}, {i, n}]], {k, 0, n-1}], {n, 1, 6}]

Prog

(PARI) for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

Crossrefs

Row sums of A333157.

Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.

Sequence in context: A111022 A086852 A084737 * A153757 A159860 A013549

Adjacent sequences: A322695 A322696 A322697 * A322699 A322700 A322701

Keyword

nonn

Author

Gus Wiseman, Dec 23 2018

Extensions

a(10)-a(18) from Andrew Howroyd, Aug 28 2019

Status

approved