This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309615 #31 Aug 12 2019 22:30:57
%S A309615 1,1,2,12,232,19230,16113300,1063117943398,225402329237199496416
%N A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.
%C A309615 First differs from A182507 at a(5) = 19230, A182507(5) = 12848.
%C A309615 A set-system is a finite set of finite nonempty sets. The dual of a set-system 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 A309615 a(n) = A326943(n) - A326944(n).
%F A309615 a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
%F A309615 a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.
%e A309615 The a(0) = 1 through a(3) = 12 set-systems:
%e A309615 {} {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
%e A309615 {{2},{1,2}} {{2},{1,2},{2,3}}
%e A309615 {{3},{1,3},{2,3}}
%e A309615 {{1},{1,2},{1,2,3}}
%e A309615 {{1},{1,3},{1,2,3}}
%e A309615 {{2},{1,2},{1,2,3}}
%e A309615 {{2},{2,3},{1,2,3}}
%e A309615 {{3},{1,3},{1,2,3}}
%e A309615 {{3},{2,3},{1,2,3}}
%e A309615 {{1},{1,2},{1,3},{1,2,3}}
%e A309615 {{2},{1,2},{2,3},{1,2,3}}
%e A309615 {{3},{1,3},{2,3},{1,2,3}}
%t A309615 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A309615 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y A309615 The version with empty edges allowed is A326943.
%Y A309615 Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.
%K A309615 nonn,more
%O A309615 0,3
%A A309615 _Gus Wiseman_, Aug 11 2019