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

%I A327078 #15 Aug 28 2019 15:36:50
%S A327078 1,1,2,8,61,969,31738,2069964,267270033,68629753641,35171000942698,
%T A327078 36024807353574280,73784587576805254653,302228602363365451957793,
%U A327078 2475873310144021668263093202,40564787336902311168400640561084
%N A327078 Binomial transform of A001187 (labeled connected graphs), if we assume A001187(1) = 0.
%C A327078 Here we consider that there is no nonempty connected graph with one vertex (different from A001187 and A182100).
%F A327078 a(n) = A182100(n) - n.
%F A327078 a(n) = A287689(n) + 1.
%e A327078 The a(0) = 1 through a(3) = 8 edge-sets:
%e A327078   {}  {}  {}       {}
%e A327078           {{1,2}}  {{1,2}}
%e A327078                    {{1,3}}
%e A327078                    {{2,3}}
%e A327078                    {{1,2},{1,3}}
%e A327078                    {{1,2},{2,3}}
%e A327078                    {{1,3},{2,3}}
%e A327078                    {{1,2},{1,3},{2,3}}
%p A327078 b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-add(
%p A327078       k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
%p A327078     end:
%p A327078 a:= n-> add(b(n-j)*binomial(n, j), j=0..n-2)+1:
%p A327078 seq(a(n), n=0..18);  # _Alois P. Heinz_, Aug 27 2019
%t A327078 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]]]]]]]]];
%t A327078 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]<=1&]],{n,0,5}]
%Y A327078 Cf. A001187, A006129, A054592, A182100, A287689, A327075.
%K A327078 nonn
%O A327078 0,3
%A A327078 _Gus Wiseman_, Aug 25 2019