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%I #6 Sep 17 2018 08:34:13
%S 1,1,0,1,1,3,3,9,12,27,42,91,151,312,550,1099,2026,3999,7527,14804,
%T 28336,55641,107737,211851,413508,814971,1600512,3162761,6241234
%N Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.
%C A rooted tree is series-reduced if every non-leaf node has at least two branches, and aperiodic if the multiplicities in the multiset of branches directly under any given node are relatively prime, and locally non-intersecting if the branches directly under any given node with more than one branch have empty intersection.
%e The a(8) = 9 rooted trees:
%e (o(o(o(o))))
%e (o(o(o)(o)))
%e (o(ooo(o)))
%e (oo(oo(o)))
%e (o(o)(o(o)))
%e (ooo(o(o)))
%e (o(o)(o)(o))
%e (ooo(o)(o))
%e (ooooo(o))
%t btrut[n_]:=btrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[btrut/@c]]]/@IntegerPartitions[n-1],And[Intersection@@#=={},GCD@@Length/@Split[#]==1]&]];
%t Table[Length[btrut[n]],{n,30}]
%Y Cf. A000081, A000837, A007562, A289509, A301700, A303431, A316470, A316473, A316475, A316495, A319270.
%K nonn,more
%O 1,6
%A _Gus Wiseman_, Sep 16 2018
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