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 A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices. 11
 1, 1, 1, 2, 4, 6, 12, 19, 35, 59, 104, 179, 318, 556, 993, 1772, 3202, 5807, 10643, 19594, 36380, 67915 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. LINKS David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014). EXAMPLE The a(1) = 1 through a(8) = 19 trees:   o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)     (ooooooo)                 (o(o))  (o(oo))   (o(ooo))   (o(oooo))    (o(ooooo))                         (oo(o))   (oo(oo))   (oo(ooo))    (oo(oooo))                         ((o)(o))  (ooo(o))   (ooo(oo))    (ooo(ooo))                                   (o(o)(o))  (oooo(o))    (oooo(oo))                                   (o(o(o)))  ((oo)(oo))   (ooooo(o))                                              (o(o(oo)))   (o(o(ooo)))                                              (o(oo(o)))   (o(oo)(oo))                                              (oo(o)(o))   (o(oo(oo)))                                              (oo(o(o)))   (o(ooo(o)))                                              ((o)(o)(o))  (oo(o(oo)))                                              (o((o)(o)))  (oo(oo(o)))                                                           (ooo(o)(o))                                                           (ooo(o(o)))                                                           (o(o)(o)(o))                                                           (o(o(o)(o)))                                                           (o(o(o(o))))                                                           (oo((o)(o)))                                                           ((o)((o)(o))) MATHEMATICA disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}]; strutsemi[n_]:=If[n==1, {{}}, If[n==2, {{{}}}, Select[Join@@Function[c, Union[Sort/@Tuples[strutsemi/@c]]]/@Rest[IntegerPartitions[n-1]], disjointQ]]]; Table[Length[strutsemi[n]], {n, 8}] CROSSREFS Not requiring lone-child-avoidance gives A316473. The non-semi version is A331680. The Matula-Goebel numbers of these trees are A331873. The same trees counted by number of leaves are A331874. Not requiring local disjointness gives A331934. Lone-child-avoiding rooted trees are A001678. Cf. A000081, A050381, A316696, A316697, A331678, A331679, A331681, A331686, A331687, A331871, A331935. Sequence in context: A094769 A068018 A294918 * A307067 A060798 A134320 Adjacent sequences:  A331869 A331870 A331871 * A331873 A331874 A331875 KEYWORD nonn,more AUTHOR Gus Wiseman, Feb 02 2020 STATUS approved

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Last modified April 17 06:46 EDT 2021. Contains 343059 sequences. (Running on oeis4.)