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A369140
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Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).
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15
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1, 1, 4, 23, 193, 2133, 29410, 486602, 9395315, 207341153, 5147194204, 141939786588, 4304047703755, 142317774817901, 5095781837539766, 196403997108015332, 8106948166404074281, 356781439557643998591, 16675999433772328981216, 824952192369049982670686
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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Inverse binomial transform of A368927.
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
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EXAMPLE
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The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
{} {{1}} {{1,2}} {{1},{2,3}}
{{1},{2}} {{2},{1,3}}
{{1},{1,2}} {{3},{1,2}}
{{2},{1,2}} {{1,2},{1,3}}
{{1,2},{2,3}}
{{1},{2},{3}}
{{1,3},{2,3}}
{{1},{2},{1,3}}
{{1},{2},{2,3}}
{{1},{3},{1,2}}
{{1},{3},{2,3}}
{{2},{3},{1,2}}
{{2},{3},{1,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{2,3}}
{{1},{1,3},{2,3}}
{{2},{1,2},{1,3}}
{{2},{1,2},{2,3}}
{{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]], {n, 0, 5}]
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PROG
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(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
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CROSSREFS
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This is the covering case of A368927.
The unlabeled version is the first differences of A369145.
A006125 counts simple graphs; also loop-graphs if shifted left.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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