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A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

%I A326909 #10 Dec 30 2019 10:06:45
%S A326909 2,2,7,45,500,9053,257151,11161244,725343385,69407094565,
%T A326909 9639771895398,1919182252611715,541764452276876719,
%U A326909 214777343584048313318,118575323291814379721651,90492591258634595795504697,94844885130660856889237907260,135738086271526574073701454370969,263921383510041055422284977248713291
%N A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.
%C A326909 Differs from A326878 in having a(0) = 2 instead of 1.
%F A326909 a(n) = A000798(n) + A006058(n). - _Jean-François Alcover_, Dec 30 2019, after _Gus Wiseman_'s comment in A006058.
%e A326909 The a(0) = 2 through a(2) = 7 sets of subsets:
%e A326909   {}    {{1}}     {{1,2}}
%e A326909   {{}}  {{},{1}}  {{},{1,2}}
%e A326909                   {{1},{1,2}}
%e A326909                   {{2},{1,2}}
%e A326909                   {{},{1},{1,2}}
%e A326909                   {{},{2},{1,2}}
%e A326909                   {{},{1},{2},{1,2}}
%t A326909 Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
%t A326909 (* Second program: *)
%t A326909 A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];
%t A326909 A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {_, _}][[All, 2]];
%t A326909 a[n_] := A006058[[n + 1]] + A000798[[n + 1]];
%t A326909 a /@ Range[0, 18] (* _Jean-François Alcover_, Dec 30 2019 *)
%Y A326909 Covering sets of subsets are A000371.
%Y A326909 The case without empty sets is A108798.
%Y A326909 The case with a single covering edge is A326878.
%Y A326909 The unlabeled version is A326898 for n > 0.
%Y A326909 The case closed only under union is A326906.
%Y A326909 The case closed only under intersection is (also) A326906.
%Y A326909 Cf. A000798, A001930, A003465, A006058, A306445, A326876, A326882, A326907, A326908.
%K A326909 nonn
%O A326909 0,1
%A A326909 _Gus Wiseman_, Aug 04 2019
%E A326909 a(18) from A000798+A006058 by _Jean-François Alcover_, Dec 30 2019