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A358588
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Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.
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13
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0, 0, 0, 0, 1, 8, 41, 171, 633, 2171, 7070, 22195, 67830, 203130, 598806, 1743258, 5023711, 14356226, 40737383, 114904941, 322432215, 900707165, 2506181060, 6948996085, 19207795836, 52944197508, 145567226556, 399314965956, 1093107693133, 2986640695436
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OFFSET
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1,6
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LINKS
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FORMULA
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a(n) = 9*a(n-1) - 32*a(n-2) + 58*a(n-3) - 58*a(n-4) + 32*a(n-5) - 9*a(n-6) + a(n-7) for n > 7.
G.f.: x^5*(-x^2 + x - 1)/((x - 1)^3*(x^2 - 3*x + 1)^2). (End)
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EXAMPLE
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The a(5) = 1 and a(6) = 8 ordered trees:
((o)(o)) ((o)(o)o)
((o)(oo))
((o)o(o))
((oo)(o))
(o(o)(o))
(((o))(o))
(((o)(o)))
((o)((o)))
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MATHEMATICA
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aot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[aot/@c], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n], Count[#, _[__], {0, Infinity}]==Depth[#]-1&]], {n, 1, 10}]
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PROG
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(PARI) \\ Needs R(n, f) defined in A358590.
seq(n) = {Vec(R(n, (h, p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023
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CROSSREFS
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For leaves instead of height we have A000891, unordered A185650 aerated.
For leaves instead of internal nodes we have A358590, unordered A358589.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
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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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