A370166 Number of unlabeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

1, 1, 3, 9, 36, 180, 1313, 14709, 277755, 9304977, 568315345, 63806703305, 13200565313255, 5042653259803433, 3567050969262370941, 4688444463558713135201, 11491940559865490367844649, 52719458629883487816297211441, 454220675869975957947658748125099

Offset: 0

Author

Gus Wiseman, Feb 12 2024

Keywords

• nonn

Examples

			Representatives of the a(0) = 1 through a(3) = 9 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{1,2},{1,3}}
             {{1},{1,2}}  {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}
                          {{1},{2},{1,3},{2,3}}
                          {{1},{1,2},{1,3},{2,3}}
		

Crossrefs

Without loops we have A002494, labeled A006129, connected A001349.

The non-covering version is A339832.

The labeled version is A370165, non-covering A079491 (apparently).

A000666 counts unlabeled loop-graphs, covering A322700.

A006125 counts labeled loop-graphs (shifted left), covering A322661.

Cf. A000085, A001187, A048291, A062740, A116539.

Programs

Mathematica

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n] && !MatchQ[#,{_,{x_},_,{y_},_,{x_,y_},_}]&]]], {n,0,4}]

Formula

First differences of A339832 (the non-covering version).