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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

Table of n, a(n) for n=1..80.

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

FORMULA

a(1) = a(2) = 1; a(n + 1) = Sum_{d|n, d<n} a(d) for n > 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.)