%I
%S 1,2,5,22,297,20536,16232437,1063231148918,225402337742595309857
%N Number of T_0 setsystems covering a subset of {1..n} that are closed under intersection.
%C A setsystem is a finite set of finite nonempty sets. The dual of a setsystem has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).
%F Binomial transform of A309615.
%e The a(0) = 1 through a(3) = 22 setsystems:
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{2}} {{2}}
%e {{1},{1,2}} {{3}}
%e {{2},{1,2}} {{1},{1,2}}
%e {{1},{1,3}}
%e {{2},{1,2}}
%e {{2},{2,3}}
%e {{3},{1,3}}
%e {{3},{2,3}}
%e {{1},{1,2},{1,3}}
%e {{2},{1,2},{2,3}}
%e {{3},{1,3},{2,3}}
%e {{1},{1,2},{1,2,3}}
%e {{1},{1,3},{1,2,3}}
%e {{2},{1,2},{1,2,3}}
%e {{2},{2,3},{1,2,3}}
%e {{3},{1,3},{1,2,3}}
%e {{3},{2,3},{1,2,3}}
%e {{1},{1,2},{1,3},{1,2,3}}
%e {{2},{1,2},{2,3},{1,2,3}}
%e {{3},{1,3},{2,3},{1,2,3}}
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y The covering case is A309615.
%Y T_0 setsystems are A326940.
%Y The version with empty edges allowed is A326945.
%Y Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 13 2019
%E a(5)a(8) from _Andrew Howroyd_, Aug 14 2019
