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%I #12 Feb 02 2024 16:11:20
%S 1,1,4,23,193,2133,29410,486602,9395315,207341153,5147194204,
%T 141939786588,4304047703755,142317774817901,5095781837539766,
%U 196403997108015332,8106948166404074281,356781439557643998591,16675999433772328981216,824952192369049982670686
%N Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).
%C These are covering loop-graphs where every connected component has a number of edges less than or equal to the number of vertices in that component. Also covering loop-graphs with at most one cycle (unicyclic) in each connected component.
%H Andrew Howroyd, <a href="/A369140/b369140.txt">Table of n, a(n) for n = 0..200</a>
%F Inverse binomial transform of A368927.
%F Exponential transform of A369197.
%F E.g.f.: exp(-x)*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(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}]],Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]],{n,0,5}]
%o (PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(-x + 3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ _Andrew Howroyd_, Feb 02 2024
%Y For a unique choice we have A000272, covering case of A088957.
%Y Without the choice condition we have A322661, unlabeled A322700.
%Y For exactly n edges we have A333331 (maybe), complement A368596.
%Y The case without loops is A367869, covering case of A133686.
%Y This is the covering case of A368927.
%Y The complement is counted by A369142, covering case of A369141.
%Y The unlabeled version is the first differences of A369145.
%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 A367862 counts graphs with n vertices and n edges, covering A367863.
%Y Cf. A000169, A000666, A003465, A006649, A062740, A116508, A137916, A368924, A368984, A369194.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 20 2024
%E a(6) onwards from _Andrew Howroyd_, Feb 02 2024
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