%I #8 Aug 12 2019 03:24:25
%S 1,2,7,112,32253,2147316942,9223372023968335715,
%T 170141183460469231667123699322514272668,
%U 5789604461865809771178549250434395393752402807429031284280914691514037561273
%N Number of connected set-systems covering a subset of {1..n}.
%C A set-system is a finite set of finite nonempty sets.
%F Binomial transform of A323818.
%e The a(0) = 1 through a(2) = 7 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%e {{1,2}}
%e {{1},{1,2}}
%e {{2},{1,2}}
%e {{1},{2},{1,2}}
%t 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 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[csm[#]]<=1&]],{n,0,4}]
%Y Covering sets of subsets are A000371.
%Y Connected graphs are A001187.
%Y The unlabeled version is A309667.
%Y The BII-numbers of connected set-systems are A326749.
%Y The covering case is A323818.
%Y Cf. A007718, A048143, A058891, A092918, A300913, A304716, A326866, A326948.
%K nonn
%O 0,2
%A _Gus Wiseman_, Aug 10 2019