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%I #7 Dec 21 2016 11:06:03
%S 1,1,0,1,0,0,1,1,0,0,2,1,2,2,2,5,4,6,8,10,14,23,26,34,46,64,81,115,
%T 158,199,277,376,505,684,934,1241,1711,2300,3123,4236,5763,7814,10647,
%U 14456,19662
%N Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.
%C A finitary set is transitive if every element is also a subset. Transitive sets are also called full sets.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Transitive_set">Transitive set</a>
%H Gus Wiseman, <a href="/A279861/a279861.png">Transitive rooted identity trees example n=23</a>
%e Sequence of transitive finitary sets begins:
%e 1 ()
%e 2 (())
%e 4 (()(()))
%e 7 (()(())((())))
%e 8 (()(())(()(())))
%e 11 (()(())((()))(((()))))
%e (()(())((()))(()(())))
%e 12 (()(())((()))(()((()))))
%e 13 (()(())((()))((())((()))))
%e (()(())(()(()))((()(()))))
%e 14 (()(())((()))(()(())((()))))
%e (()(())(()(()))(()(()(()))))
%e 15 (()(())((()))(((())))(()(())))
%e (()(())(()(()))((())(()(()))))
%e 16 (()(())((()))(((())))((((())))))
%e (()(())((()))(((())))(()((()))))
%e (()(())((()))(()(()))(()((()))))
%e (()(())((()))(()(()))((()(()))))
%e (()(())(()(()))(()(())(()(()))))
%e 17 (()(())((()))(((())))(()(((())))))
%e (()(())((()))(((())))((())((()))))
%e (()(())((()))(()(()))(()(()(()))))
%e (()(())((()))(()(()))((())((()))))
%e 18 (()(())((()))(((())))((())(((())))))
%e (()(())((()))(((())))(()(())((()))))
%e (()(())((()))(()(()))((())(()(()))))
%e (()(())((()))(()(()))(()(())((()))))
%e (()(())((()))((()((()))))(()((()))))
%e (()(())((()))(()((())))((())((()))))
%t transfins[n_]:=transfins[n]=If[n===1,{{}},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[nov,_List,{0,Infinity}]<=n]]]/@#,#]&,Array[transfins,n-1,1,Union]],Count[#,_List,{0,Infinity}]===n&]];
%t Table[Length[transfins[n]],{n,20}]
%Y Cf. A001192, A004111, A061773, A279614, A276625, A279065, A279863.
%K nonn
%O 1,11
%A _Gus Wiseman_, Dec 21 2016
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