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A358589
Number of square rooted trees with n nodes.
21
1, 0, 1, 0, 3, 2, 11, 17, 55, 107, 317, 720, 1938, 4803, 12707, 32311, 85168, 220879, 581112, 1522095, 4014186, 10568936, 27934075, 73826753, 195497427, 517927859, 1373858931, 3646158317, 9684878325, 25737819213, 68439951884, 182070121870, 484583900955, 1290213371950
OFFSET
1,5
COMMENTS
We say that a tree is square if it has the same height as number of leaves.
LINKS
EXAMPLE
The a(1) = 1 through a(7) = 11 trees:
o . (oo) . ((ooo)) ((o)(oo)) (((oooo)))
(o(oo)) (o(o)(o)) ((o(ooo)))
(oo(o)) ((oo(oo)))
((ooo(o)))
(o((ooo)))
(o(o(oo)))
(o(oo(o)))
(oo((oo)))
(oo(o(o)))
(ooo((o)))
((o)(o)(o))
MATHEMATICA
art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n], Count[#, {}, {0, Infinity}]==Depth[#]-1&]], {n, 1, 10}]
PROG
(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 + exp( sum(i=1, n-1, 1/i * subst( subst( A + O(x*x^((n-1)\i)), x, x^i), y, y^i) ) )); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h, p)->polcoef(p, h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023
CROSSREFS
For internals instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358577.
For internals instead of leaves we have A358587, ranked by A358576.
The ordered version is A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
Sequence in context: A087956 A116391 A305491 * A087629 A254214 A362995
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 23 2022
EXTENSIONS
Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved