A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

1, 2, 6, 30, 241, 2759, 40824, 736342, 15622835, 380668095, 10467815086, 320529284621, 10813165015074, 398413594789777, 15917197015926392, 685312404706694574, 31631317971844128229, 1558017329350990780607, 81567807853701988869120, 4522975947689168088308305

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

Binomial transform of A368597.

Example

The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{2}}    {{3}}
             {{1},{1,2}}  {{1},{2}}
             {{2},{1,2}}  {{1},{3}}
                          {{2},{3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Length[#]==Length[Union@@#]&]], {n, 0, 5}]

Crossrefs

The version counting all vertices is A014068.

The loopless case is A367862, counting all vertices A116508.

The covering case is A368597, connected A368951.

With inequality we have A369196, covering A369194, connected A369197.

A000085, A100861, A111924 count set partitions into singletons or pairs.

A006125 counts simple graphs, also loop-graphs if shifted left.

A006129 counts covering graphs, unlabeled A002494.

A054548 counts graphs covering n vertices with k edges, with loops A369199.

A322661 counts covering loop-graphs, unlabeled A322700.

A368927 counts choosable loop-graphs, covering A369140.

A369141 counts non-choosable loop-graphs, covering A369142.

Cf. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.

Sequence in context: A106209 A003266 A303169 * A097385 A066068 A121406

Adjacent sequences: A369195 A369196 A369197 * A369199 A369200 A369201

Keyword

nonn

Author

Gus Wiseman, Jan 18 2024

Status

approved