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A370166
Number of unlabeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.
1
1, 1, 3, 9, 36, 180, 1313, 14709, 277755, 9304977, 568315345, 63806703305, 13200565313255, 5042653259803433, 3567050969262370941, 4688444463558713135201, 11491940559865490367844649, 52719458629883487816297211441, 454220675869975957947658748125099
OFFSET
0,3
FORMULA
First differences of A339832 (the non-covering version).
EXAMPLE
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}}
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}]
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.
Sequence in context: A070960 A030834 A374662 * A030893 A030936 A030870
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2024
STATUS
approved