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 A331991 Number of semi-lone-child-avoiding achiral rooted trees with n vertices. 4
 1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 4, 4, 1, 7, 1, 7, 5, 6, 1, 7, 3, 7, 5, 7, 1, 13, 1, 8, 6, 6, 6, 10, 1, 9, 7, 9, 1, 15, 1, 12, 12, 8, 1, 12, 4, 13, 6, 11, 1, 15, 7, 13, 9, 9, 1, 17, 1, 15, 15, 9, 8, 21, 1, 13, 8, 16, 1, 18, 1, 12, 16, 11, 8, 21, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an achiral rooted tree, the branches of any given vertex are all equal. LINKS David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014). FORMULA a(1) = a(2) = 1; a(n + 1) = Sum_{d|n, d 1. G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 + x)) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, Feb 25 2020 EXAMPLE The a(n) trees for n = 2, 3, 5, 7, 11, 13:   (o)  (oo)  (oooo)    (oooooo)     (oooooooooo)        (oooooooooooo)              ((o)(o))  ((oo)(oo))   ((oooo)(oooo))      ((ooooo)(ooooo))                        ((o)(o)(o))  ((o)(o)(o)(o)(o))   ((ooo)(ooo)(ooo))                                     (((o)(o))((o)(o)))  ((oo)(oo)(oo)(oo))                                                         ((o)(o)(o)(o)(o)(o)) MATHEMATICA ab[n_]:=If[n<=2, 1, Sum[ab[d], {d, Most[Divisors[n-1]]}]]; Array[ab, 100] CROSSREFS Matula-Goebel numbers of these trees are A331992. The fully lone-child-avoiding case is A167865. The semi-achiral version is A331933. Not requiring achirality gives A331934. The identity tree version is A331964. The semi-identity tree version is A331993. Achiral rooted trees are counted by A003238. Lone-child-avoiding semi-achiral trees are A320268. Cf. A000081, A004111, A050381, A001678, A214577, A289079, A320222, A331912, A331935, A331936, A331963, A331967, A331994. Sequence in context: A124224 A290089 A323902 * A309634 A309633 A329632 Adjacent sequences:  A331988 A331989 A331990 * A331992 A331993 A331994 KEYWORD nonn AUTHOR Gus Wiseman, Feb 06 2020 STATUS approved

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Last modified April 3 04:21 EDT 2020. Contains 333195 sequences. (Running on oeis4.)