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%I #4 Mar 19 2019 07:14:41
%S 1,0,1,1,2,3,8,16,35,74,166,367,831,1878,4299,9857,22775,52777,122957,
%T 287337
%N Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.
%C An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
%C Also the number of finitary sets with n brackets where no element is also a subset. For example, the a(7) = 8 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}}}
%e The a(1) = 1 through a(8) = 16 rooted identity trees:
%e o ((o)) (((o))) ((o(o))) (((o(o)))) ((o)(o(o))) (((o))(o(o)))
%e ((((o)))) ((o((o)))) ((o(o(o)))) (((o)(o(o))))
%e (((((o))))) ((((o(o))))) (((o(o(o)))))
%e (((o)((o)))) ((o)((o(o))))
%e (((o((o))))) ((o)(o((o))))
%e ((o)(((o)))) ((o((o(o)))))
%e ((o(((o))))) ((o(o)((o))))
%e ((((((o)))))) ((o(o((o)))))
%e (((((o(o))))))
%e ((((o)((o)))))
%e ((((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],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]
%Y Cf. A000081, A290760, A304360, A306844, A317787.
%Y Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324767, A324770, A324771, A324838, A324840, A324844, A324846.
%K nonn,more
%O 1,5
%A _Gus Wiseman_, Mar 18 2019