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%I #4 Aug 22 2018 08:33:13
%S 2,2,4,8,16,32,62,122,234,451,857,1630,3068,5772,10778,20093,37259
%N Number of totally transitive rooted trees with n leaves.
%C A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root.
%e The a(5) = 16 totally transitive rooted trees with 5 leaves:
%e (o(o)(o(o)(o)))
%e (o(o)(o)(o(o)))
%e (o(o)(o)(o)(o))
%e (o(o)(oo(o)))
%e (oo(o)(o(o)))
%e (o(o)(o)(oo))
%e (oo(o)(o)(o))
%e (o(o)(ooo))
%e (o(oo)(oo))
%e (oo(o)(oo))
%e (ooo(o)(o))
%e (o(oooo))
%e (oo(ooo))
%e (ooo(oo))
%e (oooo(o))
%e (ooooo)
%t totralv[n_]:=totralv[n]=If[n==1,{{},{{}}},Join@@Table[Select[Union[Sort/@Tuples[totralv/@c]],Complement[Union@@#,#]=={}&],{c,Select[IntegerPartitions[n],Length[#]>1&]}]];
%t Table[Length[totralv[n]],{n,8}]
%Y Cf. A000081, A000669, A001678, A004111, A050381, A279861, A290689, A290760, A290822, A318185, A318186.
%K nonn,more
%O 1,1
%A _Gus Wiseman_, Aug 20 2018
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