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Number of series-reduced locally disjoint rooted trees with n unlabeled leaves.
16

%I #8 Sep 14 2018 14:33:10

%S 1,1,2,5,10,24,49,112,241,548,1218,2839,6547,15572,37179,90555,222065,

%T 552576,1384820,3506475,8936121,22941280

%N Number of series-reduced locally disjoint rooted trees with n unlabeled leaves.

%C A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

%e The a(5) = 10 trees:

%e o(o(o(oo))))

%e (o(o(ooo)))

%e (o((oo)(oo)))

%e (o(oo(oo)))

%e (o(oooo))

%e (oo(o(oo)))

%e (oo(ooo))

%e (o(oo)(oo))

%e (ooo(oo))

%e (ooooo)

%e Missing from this list but counted by A000669 are ((oo)(o(oo))) and ((oo)(ooo)).

%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];

%t nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}]];

%t Table[Length[nms[n]],{n,15}]

%Y Cf. A000081, A000669, A001678, A141268, A316471, A316471, A316475, A316495.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Jul 10 2018

%E a(18)-a(22) from _Robert Price_, Sep 14 2018