# Can this hierarchy be evaluated at higher levels?

I defined a hierarchy as follows:

$$F_{\alpha+1}(n) = \sum{^n_{m=1}} F_\alpha(m)$$, or the sum of F_a(1) + F_a(2) + ... + F_a(n)

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

The first few values of $$F_\omega$$ are 1, 2, 6, 20, 70, 252, ... (the central binomial coefficients) and the first few values of $$F_{\omega^2}$$ are 1, 4, 36, 440, 6650... but at higher levels, the hierarchy becomes very difficult to evaluate, and even $$F_{\omega^\omega}$$ gets hard to evaluate after the third value (216). It took me a whole evening last week just to evaluate $$F_{\varepsilon_0}(3) = 306030478267498284$$.

Is there an efficient way to evaluate higher-level expressions in this hierarchy?

• You are supposed to give a more specific task (e.g. calculate $F_{\varepsilon_0}(3)$), rather than asking for evaluating this hierarchy. Also, if it is not known whether there is an efficient answer, then this question would be suitable for the mathoverflow site, which is for discussion of research problems. Sep 23, 2020 at 17:17
• Some specific values I want to find: F_w^w(4), F_w^w(5), F_w^w^2(4). Very large values like F_w^w^w^2(3) (> 3^19683) and F_e_0(4) would be very hard if not impossible to exactly evaluate. Sep 23, 2020 at 19:46