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A369197
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Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.
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19
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1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)
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EXAMPLE
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The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
. {{1}} {{1,2}} {{1,2},{1,3}}
{{1},{1,2}} {{1,2},{2,3}}
{{2},{1,2}} {{1,3},{2,3}}
{{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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PROG
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(PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024
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CROSSREFS
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This is the connected case of A369194.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.
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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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