A369194 - OEIS

%I #13 Mar 23 2024 22:13:13

%S 1,1,4,23,199,2313,34015,606407,12712643,306407645,8346154699,

%T 253476928293,8490863621050,310937199521774,12356288017546937,

%U 529516578044589407,24339848939829286381,1194495870124420574751,62332449791125883072149,3446265450868329833016605

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

%C Row-sums of left portion of A369199.

%F Inverse binomial transform of A369196.

%e The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):

%e   {}  {{1}}  {{1,2}}      {{1},{2,3}}

%e              {{1},{2}}    {{2},{1,3}}

%e              {{1},{1,2}}  {{3},{1,2}}

%e              {{2},{1,2}}  {{1,2},{1,3}}

%e                           {{1,2},{2,3}}

%e                           {{1},{2},{3}}

%e                           {{1,3},{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[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

%Y The minimal case is A001862, without loops A053530.

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

%Y The case of equality is A368597, without loops A367863.

%Y The version without loops is A369191.

%Y The connected case is A369197, without loops A129271.

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

%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 A133686 counts choosable graphs, covering A367869.

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

%Y A367867 counts non-choosable graphs, covering A367868.

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

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 17 2024