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\$ to \$2n+1\$ such that no child node is less than its parent.
For example the following tree with 7 nodes is valid:
1 2 7 3 4 5 6
but the following tree is not:
1 2 7 6 4 5 3
3 is a child of
Given \$n\$ calculate the number of binary trees labeled from \$1\$ to \$2n+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.
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