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A331966
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Number of lone-child-avoiding rooted semi-identity trees with n vertices.
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14
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1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926
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OFFSET
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1,5
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COMMENTS
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Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
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EXAMPLE
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The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)
(o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))
(oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))
(ooo(oo)) (ooo(ooo)) (ooo(oooo))
(o(o(oo))) (oooo(oo)) (oooo(ooo))
((oo)(ooo)) (ooooo(oo))
(o(o(ooo))) ((oo)(oooo))
(o(oo(oo))) (o(o(oooo)))
(oo(o(oo))) (o(oo)(ooo))
(o(oo(ooo)))
(o(ooo(oo)))
(oo(o(ooo)))
(oo(oo(oo)))
(ooo(o(oo)))
((oo)(o(oo)))
(o(o(o(oo))))
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MATHEMATICA
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ssb[n_]:=If[n==1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[ssb/@c]], UnsameQ@@DeleteCases[#, {}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]], {n, 10}]
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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=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020
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CROSSREFS
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The non-semi case is A000007.
Lone-child-avoiding rooted trees are A001678.
The locally disjoint case is A212804.
Not requiring lone-child-avoidance gives A306200.
Matula-Goebel numbers of these trees are A331965.
The semi-lone-child-avoiding version is A331993.
Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.
Sequence in context: A335703 A107250 A050168 * A072176 A329700 A217282
Adjacent sequences: A331963 A331964 A331965 * A331967 A331968 A331969
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Feb 05 2020
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EXTENSIONS
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Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020
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STATUS
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approved
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