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%I #13 Aug 13 2021 07:51:25
%S 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,
%T 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,
%U 15,9,8,21,1,13,8,16,1,18,1,12,16,11,8,21,1
%N Number of semi-lone-child-avoiding achiral rooted trees with n vertices.
%C A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
%C In an achiral rooted tree, the branches of any given vertex are all equal.
%H Seiichi Manyama, <a href="/A331991/b331991.txt">Table of n, a(n) for n = 1..10000</a>
%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>
%F a(1) = a(2) = 1; a(n + 1) = Sum_{d|n, d<n} a(d) for n > 1.
%F G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 + x)) * Sum_{k>=1} A(x^k)). - _Ilya Gutkovskiy_, Feb 25 2020
%e The a(n) trees for n = 2, 3, 5, 7, 11, 13:
%e (o) (oo) (oooo) (oooooo) (oooooooooo) (oooooooooooo)
%e ((o)(o)) ((oo)(oo)) ((oooo)(oooo)) ((ooooo)(ooooo))
%e ((o)(o)(o)) ((o)(o)(o)(o)(o)) ((ooo)(ooo)(ooo))
%e (((o)(o))((o)(o))) ((oo)(oo)(oo)(oo))
%e ((o)(o)(o)(o)(o)(o))
%t ab[n_]:=If[n<=2,1,Sum[ab[d],{d,Most[Divisors[n-1]]}]];
%t Array[ab,100]
%Y Matula-Goebel numbers of these trees are A331992.
%Y The fully lone-child-avoiding case is A167865.
%Y The semi-achiral version is A331933.
%Y Not requiring achirality gives A331934.
%Y The identity tree version is A331964.
%Y The semi-identity tree version is A331993.
%Y Achiral rooted trees are counted by A003238.
%Y Lone-child-avoiding semi-achiral trees are A320268.
%Y Cf. A000081, A004111, A050381, A001678, A214577, A289079, A320222, A331912, A331935, A331936, A331963, A331967, A331994.
%K nonn
%O 1,5
%A _Gus Wiseman_, Feb 06 2020
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