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A370318
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Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.
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5
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0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363
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OFFSET
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0,5
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COMMENTS
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The case of an empty edge set is excluded.
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LINKS
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FORMULA
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Binomial transform of A057500 (if the null graph is not connected).
a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!). - Andrew Howroyd, Feb 19 2024
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MATHEMATICA
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csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[#]==Length[Union@@#] && Length[csm[#]]==1&]], {n, 0, 5}]
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PROG
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a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024
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CROSSREFS
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The covering case is A057500, which is also the covering case of A370317.
A062734 counts connected graphs by edge count.
A143543 counts simple labeled graphs by number of connected components.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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