A368927 - OEIS

%I #12 Feb 02 2024 16:11:16

%S 1,2,7,39,314,3374,45630,744917,14245978,312182262,7708544246,

%T 211688132465,6397720048692,210975024924386,7537162523676076,

%U 289952739051570639,11949100971787370300,525142845422124145682,24515591201199758681892,1211486045654016217202663

%N 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 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 Andrew Howroyd, <a href="/A368927/b368927.txt">Table of n, a(n) for n = 0..200</a>

%F Binomial transform of A369140.

%F Exponential transform of A369197 with A369197(1) = 2.

%F 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 The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):

%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[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}]

%o (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 Without the choice condition we have A006125.

%Y The case of a unique choice is A088957, unlabeled A087803.

%Y The case without loops is A133686, complement A367867, covering A367869.

%Y For exactly n edges and no loops we have A137916, unlabeled A137917.

%Y For exactly n edges we have A333331 (maybe), complement A368596.

%Y For edges of any positive size we have A367902, complement A367903.

%Y The covering case is A369140, complement A369142.

%Y The complement is counted by A369141.

%Y The complement for n edges and no loops is A369143, covering A369144.

%Y The unlabeled version is A369145, complement A369146.

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

%Y A006129 counts covering graphs, unlabeled A002494.

%Y A322661 counts labeled covering loop-graphs, connected A062740.

%Y Cf. A000169, A000272, A000666, A014068, A057500, A116508, A367863, A368601, A368924, A368984, A369200.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jan 15 2024

%E a(7) onwards from _Andrew Howroyd_, Feb 02 2024