%I #9 Jan 18 2024 04:40:49

%S 1,2,7,39,320,3584,51405,900947,18661186,445827942,12062839691,

%T 364451604095,12157649050827,443713171974080,17583351295466338,

%U 751745326170662049,34485624653535808340,1689485711682987916502,88030098291829749593643,4860631073631586486397141

%N Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

%F Binomial transform of A369194.

%e The a(0) = 1 through a(2) = 7 loop-graphs:

%e {} {} {}

%e {{1}} {{1}}

%e {{2}}

%e {{1,2}}

%e {{1},{2}}

%e {{1},{1,2}}

%e {{2},{1,2}}

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

%Y The version counting all vertices is A066383, without loops A369192.

%Y The loopless case is A369193, with case of equality A367862.

%Y The covering case is A369194, connected A369197, minimal case A001862.

%Y The case of equality is A369198, covering case A368597.

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

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

%Y A006129 counts covering graphs, unlabeled A002494.

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

%Y A322661 counts covering loop-graphs, unlabeled A322700.

%Y A368927 counts choosable loop-graphs, covering A369140.

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

%Y Cf. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jan 17 2024