How to calculate this hierarchy in Mathematica?

A hierarchy can be defined as follows:

$$F_0(n) = 1$$ when n = 1, 0 for all other values

$$F_{\alpha+1}(n) = \sum_{i=1}^nF_\alpha(i)$$

$$F_{\alpha}(n) = F_{\alpha[n]}(n)$$ when $$\alpha$$ is a limit ordinal

For the last part, I use the most common system of fundamental sequences, where the n-th term of the fundamental sequence of $$\omega$$ is n, the n-th term for $$\omega*(\alpha+1)$$ is w*a + n, the n-th term for w^2 is $$\omega \times n$$, and the n-th term for $$\omega^{\alpha+1}$$ is $$\omega^\alpha \times n$$.

The functions below omega are the binomial coefficients, and $$F_\omega(n)$$ is the central binomial coefficients. However, the functions after that lack clean formulas. The only way to evaluate them as far as I know is by recursively adding up the values of all the previous functions.

Is it possible to write a Mathematica code to evaluate these functions? (Most wanted: $$F_{\omega^2}(n)$$, $$F_{\omega^\omega}(n)$$, although it would be even better if we could program any function in the linear/polynomial omega level)

Even better, can it be done without taking large amounts of memory/time for relatively small inputs (especially when evaluating higher-level functions such as F_w^w(n))?