A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Author

Gus Wiseman, Jan 17 2024

Keywords

• nonn

Comments

Row-sums of left portion of A369199.

Examples

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{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}}
		

Crossrefs

The minimal case is A001862, without loops A053530.

This is the covering case of A066383 and A369196, cf. A369192 and A369193.

The case of equality is A368597, without loops A367863.

The version without loops is A369191.

The connected case is A369197, without loops A129271.

The unlabeled version is A370169, equality A368599, non-covering A368598.

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.

A133686 counts choosable graphs, covering A367869.

A322661 counts covering loop-graphs, unlabeled A322700.

A367867 counts non-choosable graphs, covering A367868.

A368927 counts choosable loop-graphs, covering A369140.

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

Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Programs

Mathematica

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

Formula

Inverse binomial transform of A369196.