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
Seiichi Manyama, Table of n, a(n) for n = 1..10000
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<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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 06 2020
STATUS
approved