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Increasing permutation trees

For this challenge a "binary tree" is a rooted tree where each node has 0 children (leaf) or 2. The children of a node are unordered, meaning that while you might draw the tree with left and right children there isn't a distinction between them and mirroring the tree or a sub-tree does not produce a new tree.

In this challenge you will be given an integer \$n\$ and you will be asked to determine the number of binary trees with each node labeled from \1ドル\$ to \2ドルn+1\$ such that no child node is less than its parent.

For example the following tree with 7 nodes is valid:

but the following tree is not:

Since 3 is a child of 4.

Task

Given \$n\$ calculate the number of binary trees labeled from \1ドル\$ to \2ドルn+1\$ such that no child node is less than its parent.

This is code-golf so the goal is to minimize the size of your source code as measured in bytes.

Test cases

I've calculated the first 11 solutions on my own:

1, 1, 4, 34, 496, 11056, 349504, 14873104, 819786496, 56814228736, 4835447317504

and the OEIS has more terms at: A002105

Answer*

If this is an answer to a challenge…

…Be sure to follow the challenge specification. However, please refrain from exploiting obvious loopholes. Answers abusing any of the standard loopholes are considered invalid. If you think a specification is unclear or underspecified, comment on the question instead.
…Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one. Explanations of your answer make it more interesting to read and are very much encouraged.
…Include a short header which indicates the language(s) of your code and its score, as defined by the challenge.

More generally…

…Please make sure to answer the question and provide sufficient detail.
…Avoid asking for help, clarification or responding to other answers (use comments instead).

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\$\begingroup\$ for _ in range(n+n-2) -> for _ in"_"*~-n*2 for 81 bytes: Try it online! \$\endgroup\$

pxeger
– pxeger

2022年06月04日 19:32:30 +00:00
Commented Jun 4, 2022 at 19:32
3

\$\begingroup\$ @pxeger Even I know a golfier way of saying "_"*2... \$\endgroup\$

Neil
– Neil

2022年06月04日 21:45:35 +00:00
Commented Jun 4, 2022 at 21:45
\$\begingroup\$ Heh, good point. \$\endgroup\$

pxeger
– pxeger

2022年06月05日 06:04:42 +00:00
Commented Jun 5, 2022 at 6:04

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