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%I A331874 #7 Feb 03 2020 22:18:15
%S A331874 2,3,8,24,67,214,687,2406,8672,32641,125431,493039,1964611
%N A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.
%C A331874 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
%C A331874 Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
%H A331874 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A331874 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e A331874 The a(1) = 2 through a(4) = 24 trees:
%e A331874   o    (oo)      (ooo)          (oooo)
%e A331874   (o)  (o(o))    (o(oo))        (o(ooo))
%e A331874        ((o)(o))  (oo(o))        (oo(oo))
%e A331874                  (o(o)(o))      (ooo(o))
%e A331874                  (o(o(o)))      ((oo)(oo))
%e A331874                  ((o)(o)(o))    (o(o(oo)))
%e A331874                  (o((o)(o)))    (o(oo(o)))
%e A331874                  ((o)((o)(o)))  (oo(o)(o))
%e A331874                                 (oo(o(o)))
%e A331874                                 (o(o)(o)(o))
%e A331874                                 (o(o(o)(o)))
%e A331874                                 (o(o(o(o))))
%e A331874                                 (oo((o)(o)))
%e A331874                                 ((o)(o)(o)(o))
%e A331874                                 ((o(o))(o(o)))
%e A331874                                 ((oo)((o)(o)))
%e A331874                                 (o((o)(o)(o)))
%e A331874                                 (o(o)((o)(o)))
%e A331874                                 (o(o((o)(o))))
%e A331874                                 ((o)((o)(o)(o)))
%e A331874                                 ((o)(o)((o)(o)))
%e A331874                                 (o((o)((o)(o))))
%e A331874                                 (((o)(o))((o)(o)))
%e A331874                                 ((o)((o)((o)(o))))
%t A331874 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A331874 slaurt[n_]:=If[n==1,{o,{o}},Join@@Table[Select[Union[Sort/@Tuples[slaurt/@ptn]],disjointQ[Select[#,!AtomQ[#]&]]&],{ptn,Rest[IntegerPartitions[n]]}]];
%t A331874 Table[Length[slaurt[n]],{n,8}]
%Y A331874 Not requiring local disjointness gives A050381.
%Y A331874 The non-semi version is A316697.
%Y A331874 The same trees counted by number of vertices are A331872.
%Y A331874 The Matula-Goebel numbers of these trees are A331873.
%Y A331874 Lone-child-avoiding rooted trees counted  by leaves are A000669.
%Y A331874 Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
%Y A331874 Cf. A000081, A001678, A300660, A316473, A316495, A316696, A331678, A331679, A331680, A331682, A331687, A331871, A331935.
%K A331874 nonn,more
%O A331874 1,1
%A A331874 _Gus Wiseman_, Feb 02 2020

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