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%I A322137 #11 Sep 13 2021 20:59:18
%S A322137 1,1,3,17,140,1524,20673,336259,6382302,138525780,3384988809,
%T A322137 91976158434,2751122721402,89833276321440,3179852538140115,
%U A322137 121287919647418118,4959343701136929850,216406753768138678671,10037782414506891597734,493175891246093032826160
%N A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).
%H A322137 Andrew Howroyd, <a href="/A322137/b322137.txt">Table of n, a(n) for n = 0..200</a>
%H A322137 P. S. Kolesnikov and B. K. Sartayev, <a href="https://arxiv.org/abs/2105.13815">On the special identities of Gelfand--Dorfman algebras</a>, arXiv:2105.13815 [math.RA], 2021.
%H A322137 Gus Wiseman, <a href="/A322137/a322137.png">The a(4) = 140 labeled connected graphs with 4 edges.</a>
%t A322137 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A322137 Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]
%o A322137 (PARI)
%o A322137 Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
%o A322137 seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ _Andrew Howroyd_, Nov 28 2018
%Y A322137 Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.
%K A322137 nonn
%O A322137 0,3
%A A322137 _Gus Wiseman_, Nov 27 2018
%E A322137 Terms a(8) and beyond from _Andrew Howroyd_, Nov 28 2018

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