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))?



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