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%I #11 Jun 20 2020 02:04:59
%S 1,1,1,1,3,4,9,20,41,89,196,443,987,2246,5114,11757,27122,62898,
%T 146392,342204,802429,1887882
%N Number of anti-transitive rooted identity trees with n nodes.
%C A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
%C Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (o = {}):
%C {{{{{{{o}}}}}}}
%C {{{{{o,{o}}}}}}
%C {{{{o,{{o}}}}}}
%C {{{o,{{{o}}}}}}
%C {{{o,{o,{o}}}}}
%C {{{{o},{{o}}}}}
%C {{o,{{{{o}}}}}}
%C {{o,{{o,{o}}}}}
%C {{o,{o,{{o}}}}}
%C {{{o},{{{o}}}}}
%C {{{o},{o,{o}}}}
%C {{o,{o},{{o}}}}
%C {o,{{{{{o}}}}}}
%C {o,{{{o,{o}}}}}
%C {o,{{o,{{o}}}}}
%C {o,{{o},{{o}}}}
%C {{o},{{{{o}}}}}
%C {{o},{{o,{o}}}}
%C {{o},{o,{{o}}}}
%C {{{o}},{o,{o}}}
%H Gus Wiseman, <a href="/A324764/a324764.png">The a(9) = 41 anti-transitive rooted identity trees</a>.
%e The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
%e o (o) ((o)) (((o))) ((o(o))) (((o(o)))) ((o(o(o))))
%e (o((o))) ((o((o)))) (o((o(o))))
%e ((((o)))) (o(((o)))) ((((o(o)))))
%e (((((o))))) (((o)((o))))
%e (((o((o)))))
%e ((o)(((o))))
%e ((o(((o)))))
%e (o((((o)))))
%e ((((((o))))))
%t idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
%t Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]
%Y Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
%Y Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.
%K nonn,more
%O 1,5
%A _Gus Wiseman_, Mar 17 2019
%E a(21)-a(22) from _Jinyuan Wang_, Jun 20 2020
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