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%I #8 Jan 20 2024 09:17:36
%S 1,2,6,30,241,2759,40824,736342,15622835,380668095,10467815086,
%T 320529284621,10813165015074,398413594789777,15917197015926392,
%U 685312404706694574,31631317971844128229,1558017329350990780607,81567807853701988869120,4522975947689168088308305
%N Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.
%F Binomial transform of A368597.
%e The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{2}} {{2}}
%e {{1},{2}} {{3}}
%e {{1},{1,2}} {{1},{2}}
%e {{2},{1,2}} {{1},{3}}
%e {{2},{3}}
%e {{1},{1,2}}
%e {{1},{1,3}}
%e {{2},{1,2}}
%e {{2},{2,3}}
%e {{3},{1,3}}
%e {{3},{2,3}}
%e {{1},{2},{3}}
%e {{1},{2},{1,3}}
%e {{1},{2},{2,3}}
%e {{1},{3},{1,2}}
%e {{1},{3},{2,3}}
%e {{2},{3},{1,2}}
%e {{2},{3},{1,3}}
%e {{1},{1,2},{1,3}}
%e {{1},{1,2},{2,3}}
%e {{1},{1,3},{2,3}}
%e {{2},{1,2},{1,3}}
%e {{2},{1,2},{2,3}}
%e {{2},{1,3},{2,3}}
%e {{3},{1,2},{1,3}}
%e {{3},{1,2},{2,3}}
%e {{3},{1,3},{2,3}}
%e {{1,2},{1,3},{2,3}}
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]
%Y The version counting all vertices is A014068.
%Y The loopless case is A367862, counting all vertices A116508.
%Y The covering case is A368597, connected A368951.
%Y With inequality we have A369196, covering A369194, connected A369197.
%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. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.
%K nonn
%O 0,2
%A _Gus Wiseman_, Jan 18 2024
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