%I #15 Feb 01 2020 07:08:24

%S 1,0,1,1,2,3,6,9,16,26,45,72,124,201,341,561,947,1571,2651,4434,7496,

%T 12631,21423,36332,61910,105641,180924,310548,534713,923047

%N Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

%C First differs from A320268 at a(11) = 45, A320268(11) = 44.

%C A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

%H 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 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 The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):

%e o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)

%e (o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))

%e (oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))

%e (ooo(oo)) (ooo(ooo)) (ooo(oooo))

%e ((oo)(oo)) (oooo(oo)) (oooo(ooo))

%e (o(o(oo))) (o(o(ooo))) (ooooo(oo))

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

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

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

%e (o(ooo(oo)))

%e (oo(o(ooo)))

%e (oo(oo)(oo))

%e (oo(oo(oo)))

%e (ooo(o(oo)))

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

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

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

%t strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];

%t Table[Length[strut[n]],{n,10}]

%Y The enriched version is A316696.

%Y The Matula-Goebel numbers of these trees are A331871.

%Y The non-locally disjoint version is A001678.

%Y These trees counted by number of leaves are A316697.

%Y The semi-lone-child-avoiding version is A331872.

%Y Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

%K nonn,more

%O 1,5

%A _Gus Wiseman_, Jan 25 2020