cp's OEIS Frontend

A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

%I A368927 #12 Feb 02 2024 16:11:16
%S A368927 1,2,7,39,314,3374,45630,744917,14245978,312182262,7708544246,
%T A368927 211688132465,6397720048692,210975024924386,7537162523676076,
%U A368927 289952739051570639,11949100971787370300,525142845422124145682,24515591201199758681892,1211486045654016217202663
%N A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.
%C A368927 These are loop-graphs where every connected component has a number of edges less than or equal to the number of vertices. Also loop-graphs with at most one cycle (unicyclic) in each connected component.
%H A368927 Andrew Howroyd, <a href="/A368927/b368927.txt">Table of n, a(n) for n = 0..200</a>
%F A368927 Binomial transform of A369140.
%F A368927 Exponential transform of A369197 with A369197(1) = 2.
%F A368927 E.g.f.: exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - _Andrew Howroyd_, Feb 02 2024
%e A368927 The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
%e A368927   {}  {}     {}
%e A368927       {{1}}  {{1}}
%e A368927              {{2}}
%e A368927              {{1,2}}
%e A368927              {{1},{2}}
%e A368927              {{1},{1,2}}
%e A368927              {{2},{1,2}}
%t A368927 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}]
%o A368927 (PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ _Andrew Howroyd_, Feb 02 2024
%Y A368927 Without the choice condition we have A006125.
%Y A368927 The case of a unique choice is A088957, unlabeled A087803.
%Y A368927 The case without loops is A133686, complement A367867, covering A367869.
%Y A368927 For exactly n edges and no loops we have A137916, unlabeled A137917.
%Y A368927 For exactly n edges we have A333331 (maybe), complement A368596.
%Y A368927 For edges of any positive size we have A367902, complement A367903.
%Y A368927 The covering case is A369140, complement A369142.
%Y A368927 The complement is counted by A369141.
%Y A368927 The complement for n edges and no loops is A369143, covering A369144.
%Y A368927 The unlabeled version is A369145, complement A369146.
%Y A368927 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A368927 A006129 counts covering graphs, unlabeled A002494.
%Y A368927 A322661 counts labeled covering loop-graphs, connected A062740.
%Y A368927 Cf. A000169, A000272, A000666, A014068, A057500, A116508, A367863, A368601, A368924, A368984, A369200.
%K A368927 nonn
%O A368927 0,2
%A A368927 _Gus Wiseman_, Jan 15 2024
%E A368927 a(7) onwards from _Andrew Howroyd_, Feb 02 2024