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A331991
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Number of semi-lone-child-avoiding achiral rooted trees with n vertices.
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5
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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
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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))
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MATHEMATICA
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ab[n_]:=If[n<=2, 1, Sum[ab[d], {d, Most[Divisors[n-1]]}]];
Array[ab, 100]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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