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%I A367863 #17 Dec 29 2023 16:41:09
%S A367863 1,0,0,1,15,222,3760,73755,1657845,42143500,1197163134,37613828070,
%T A367863 1295741321875,48577055308320,1969293264235635,85852853154670693,
%U A367863 4005625283891276535,199166987259400191480,10513996906985414443720,587316057411626070658200,34612299496604684775762261
%N A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.
%H A367863 Andrew Howroyd, <a href="/A367863/b367863.txt">Table of n, a(n) for n = 0..200</a>
%F A367863 Binomial transform is A367862.
%F A367863 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n). - _Andrew Howroyd_, Dec 29 2023
%e A367863 Non-isomorphic representatives of the a(4) = 15 graphs:
%e A367863   {{1,2},{1,3},{1,4},{2,3}}
%e A367863   {{1,2},{1,3},{2,4},{3,4}}
%t A367863 Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[#]==n&]],{n,0,5}]
%o A367863 (PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n)) \\ _Andrew Howroyd_, Dec 29 2023
%Y A367863 The connected case is A057500, unlabeled A001429.
%Y A367863 The unlabeled version is A006649.
%Y A367863 The non-covering version is A116508.
%Y A367863 For set-systems we have A367916, ranks A367917.
%Y A367863 A001187 counts connected graphs, A001349 unlabeled.
%Y A367863 A006125 counts graphs, A000088 unlabeled.
%Y A367863 A006129 counts covering graphs, A002494 unlabeled.
%Y A367863 A058891 counts set-systems, unlabeled A000612, without singletons A016031.
%Y A367863 A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
%Y A367863 A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
%Y A367863 A143543 counts simple labeled graphs by number of connected components.
%Y A367863 A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
%Y A367863 A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
%Y A367863 Cf. A003465, A006126, A305000, A316983, A319559, A323817, A326754, A367769, A367901, A367902, A367903.
%K A367863 nonn
%O A367863 0,5
%A A367863 _Gus Wiseman_, Dec 07 2023
%E A367863 Terms a(8) and beyond from _Andrew Howroyd_, Dec 29 2023

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