A327078 Binomial transform of A001187 (labeled connected graphs), if we assume A001187(1) = 0.

1, 1, 2, 8, 61, 969, 31738, 2069964, 267270033, 68629753641, 35171000942698, 36024807353574280, 73784587576805254653, 302228602363365451957793, 2475873310144021668263093202, 40564787336902311168400640561084

Offset: 0

Author

Gus Wiseman, Aug 25 2019

Keywords

• nonn

Comments

Here we consider that there is no nonempty connected graph with one vertex (different from A001187 and A182100).

Examples

			The a(0) = 1 through a(3) = 8 edge-sets:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}
		

Crossrefs

Cf. A001187, A006129, A054592, A182100, A287689, A327075.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-add(
      k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
a:= n-> add(b(n-j)*binomial(n, j), j=0..n-2)+1:
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 27 2019

Mathematica

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,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[csm[#]]<=1&]],{n,0,5}]

Formula

a(n) = A182100(n) - n.

a(n) = A287689(n) + 1.