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A331964
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Number of semi-lone-child-avoiding rooted identity trees with n vertices.
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12
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1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 44, 74, 123, 209, 353, 602, 1026, 1760, 3019, 5203, 8977, 15538, 26930, 46792, 81415, 141939, 247795, 433307, 758672, 1330219, 2335086, 4104064, 7220937, 12718694, 22424283, 39574443, 69903759, 123584852, 218668323
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OFFSET
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1,8
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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. It is an identity tree if the branches of any given vertex are all distinct.
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LINKS
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EXAMPLE
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The a(9) = 2 through a(12) = 10 semi-lone-child-avoiding rooted identity trees:
((o)(o(o(o)))) (o(o)(o(o(o)))) ((o)(o(o)(o(o)))) (o(o)(o(o)(o(o))))
(o((o)(o(o)))) (o(o(o)(o(o)))) ((o)(o(o(o(o))))) (o(o)(o(o(o(o)))))
(o(o(o(o(o))))) ((o(o))(o(o(o)))) (o(o(o))(o(o(o))))
((o)((o)(o(o)))) (o((o)(o(o(o))))) (o(o(o)(o(o(o)))))
(o(o)((o)(o(o)))) (o(o(o(o)(o(o)))))
(o(o((o)(o(o))))) (o(o(o(o(o(o))))))
((o)((o)(o(o(o)))))
((o)(o((o)(o(o)))))
((o(o))((o)(o(o))))
(o((o)((o)(o(o)))))
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MATHEMATICA
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ssei[n_]:=Switch[n, 1, {{}}, 2, {{{}}}, _, Join@@Function[c, Select[Union[Sort/@Tuples[ssei/@c]], UnsameQ@@#&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssei[n]], {n, 15}]
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PROG
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(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=[1, 1]); for(n=2, n-1, v=concat(v, WeighT(v)[n] - v[n])); v} \\ Andrew Howroyd, Feb 09 2020
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CROSSREFS
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Matula-Goebel numbers of these trees are A331963.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A050381, A276625, A300660, A306200, A316694, A331872, A331875, A331933, A331935, A331936, A331937, A331966.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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