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%I #22 Nov 20 2023 11:39:40
%S 1,2,7,45,500,9053,257151,11161244,725343385,69407094565,
%T 9639771895398,1919182252611715,541764452276876719,
%U 214777343584048313318,118575323291814379721651,90492591258634595795504697,94844885130660856889237907260,135738086271526574073701454370969,263921383510041055422284977248713291
%N Number of topologies whose points are a subset of {1..n}.
%H Wikipedia <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</a>
%F From _Geoffrey Critzer_, Jul 12 2022: (Start)
%F E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for A001035.
%F a(n) = Sum_{k=0..n} binomial(n,k)*A000798(k). (End)
%e The a(0) = 1 through a(2) = 7 topologies:
%e {{}} {{}} {{}}
%e {{},{1}} {{},{1}}
%e {{},{2}}
%e {{},{1,2}}
%e {{},{1},{1,2}}
%e {{},{2},{1,2}}
%e {{},{1},{2},{1,2}}
%t Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}]
%t (* Second program: *)
%t A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];
%t a[n_] := Sum[Binomial[n, k]*A000798[[k+1]], {k, 0, n}];
%t a /@ Range[0, Length[A000798]-1] (* _Jean-François Alcover_, Dec 30 2019 *)
%Y Binomial transform of A000798 (the covering case).
%Y Cf. A001035, A001930, A003465, A014466, A102896, A102897, A306445, A326866, A326876.
%K nonn
%O 0,2
%A _Gus Wiseman_, Jul 30 2019
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