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A358590
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Number of square ordered rooted trees with n nodes.
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18
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1, 0, 1, 0, 6, 5, 36, 84, 309, 890, 3163, 9835, 32979, 108252, 360696, 1192410, 3984552, 13276769, 44371368, 148402665, 497072593, 1665557619, 5586863093, 18750662066, 62968243731, 211565969511, 711187790166, 2391640404772, 8045964959333, 27077856222546
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OFFSET
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1,5
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COMMENTS
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We say that a tree is square if it has the same height as number of leaves.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(6) = 5 ordered trees:
o . (oo) . ((o)oo) ((o)(o)o)
((oo)o) ((o)(oo))
((ooo)) ((o)o(o))
(o(o)o) ((oo)(o))
(o(oo)) (o(o)(o))
(oo(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) \\ R(n, f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n, f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1 + 1/(1 - A + O(x^n))); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h, p)->polcoef(p, h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023
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CROSSREFS
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For internals instead of height we have A000891, unordered A185650 aerated.
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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