There is a non recursive formula for T(x,k)?

[marraco]

Because T(x,k) counts "number of forests of labeled rooted trees with x nodes and height at most k", to extend tetration to rational exponents, we need a working definition of "(rooted labeled) tree with non integer height x".

Here is an example of a rooted labeled tree:

The height x=4, because from the root the most distant node is 4 nodes away.
(the root is also labeled 4, but that's an accident and irrelevant)

So, one way we can define what is a tree with non integer height, is to allow the duplication of labels.
This tree has 9 nodes wit labels {1,2,3,4,5,6,7,8,9}. If we allow duplication of labels, we will have 18 nodes labeled {1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9}

We can agree that n copies of a label g cause the label to hold a weight of 1/n

For example, if we allow 2 copies of each label, the former tree would have a height 2, composed of half of the nodes labeled {3,8,5,2} (I'm not sure if the root 4 should be counted)

Because now we have 18 nodes, the number of forests is much larger, but we need an isomorphism (a bijection) between trees with duplicated labels, and trees with integer labels, so the count T(x,k) returns the same number of forests, for 9 integer nodes, and isomorphic trees with 18 half nodes.

What we need are additional restrictions on the way trees are allowed to be made with fractional nodes, or additional rules on how to count fractional trees and forests.