%I #17 Dec 29 2023 16:41:09

%S 1,0,0,1,15,222,3760,73755,1657845,42143500,1197163134,37613828070,

%T 1295741321875,48577055308320,1969293264235635,85852853154670693,

%U 4005625283891276535,199166987259400191480,10513996906985414443720,587316057411626070658200,34612299496604684775762261

%N Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

%H Andrew Howroyd, <a href="/A367863/b367863.txt">Table of n, a(n) for n = 0..200</a>

%F Binomial transform is A367862.

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n). - _Andrew Howroyd_, Dec 29 2023

%e Non-isomorphic representatives of the a(4) = 15 graphs:

%e {{1,2},{1,3},{1,4},{2,3}}

%e {{1,2},{1,3},{2,4},{3,4}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[#]==n&]],{n,0,5}]

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n)) \\ _Andrew Howroyd_, Dec 29 2023

%Y The connected case is A057500, unlabeled A001429.

%Y The unlabeled version is A006649.

%Y The non-covering version is A116508.

%Y For set-systems we have A367916, ranks A367917.

%Y A001187 counts connected graphs, A001349 unlabeled.

%Y A006125 counts graphs, A000088 unlabeled.

%Y A006129 counts covering graphs, A002494 unlabeled.

%Y A058891 counts set-systems, unlabeled A000612, without singletons A016031.

%Y A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.

%Y A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.

%Y A143543 counts simple labeled graphs by number of connected components.

%Y A323818 counts connected set-systems, unlabeled A323819, ranks A326749.

%Y A367867 = graphs contradict strict AoC, connected A140638, covering A367868.

%Y Cf. A003465, A006126, A305000, A316983, A319559, A323817, A326754, A367769, A367901, A367902, A367903.

%K nonn

%O 0,5

%A _Gus Wiseman_, Dec 07 2023

%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 29 2023