cp's OEIS Frontend

A326878 Number of topologies whose points are a subset of {1..n}.

%I A326878 #22 Nov 20 2023 11:39:40
%S A326878 1,2,7,45,500,9053,257151,11161244,725343385,69407094565,
%T A326878 9639771895398,1919182252611715,541764452276876719,
%U A326878 214777343584048313318,118575323291814379721651,90492591258634595795504697,94844885130660856889237907260,135738086271526574073701454370969,263921383510041055422284977248713291
%N A326878 Number of topologies whose points are a subset of {1..n}.
%H A326878 Wikipedia <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</a>
%F A326878 From _Geoffrey Critzer_, Jul 12 2022: (Start)
%F A326878 E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for A001035.
%F A326878 a(n) = Sum_{k=0..n} binomial(n,k)*A000798(k). (End)
%e A326878 The a(0) = 1 through a(2) = 7 topologies:
%e A326878   {{}}  {{}}      {{}}
%e A326878         {{},{1}}  {{},{1}}
%e A326878                   {{},{2}}
%e A326878                   {{},{1,2}}
%e A326878                   {{},{1},{1,2}}
%e A326878                   {{},{2},{1,2}}
%e A326878                   {{},{1},{2},{1,2}}
%t A326878 Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}]
%t A326878 (* Second program: *)
%t A326878 A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];
%t A326878 a[n_] := Sum[Binomial[n, k]*A000798[[k+1]], {k, 0, n}];
%t A326878 a /@ Range[0, Length[A000798]-1] (* _Jean-François Alcover_, Dec 30 2019 *)
%Y A326878 Binomial transform of A000798 (the covering case).
%Y A326878 Cf. A001035, A001930, A003465, A014466, A102896, A102897, A306445, A326866, A326876.
%K A326878 nonn
%O A326878 0,2
%A A326878 _Gus Wiseman_, Jul 30 2019