A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.

Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Mathematica

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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1], {2}], {n}], And[Union@@#==Range[Max@@Union@@#], Length[csm[#]]==1]&]], {n, 6}]

Prog

(PARI)
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}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j, 2)))))} \\ Andrew Howroyd, Nov 28 2018

Crossrefs

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

Sequence in context: A025167 A136727 A291842 * A062873 A120022 A001865

Adjacent sequences: A322134 A322135 A322136 * A322138 A322139 A322140

Keyword

nonn

Author

Gus Wiseman, Nov 27 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

Status

approved