The equation in question:
$$f^{[s]}(x)=\lim_{n\to\infty}\binom sn\sum_{k=0}^n\frac{s-n}{s-k}\binom nk(-1)^{n-k}f^{[k]}(x)$$
where $f^{[s]}(x)$ is the $s$th iteration of $f(x)$.
This formula is mentioned here along with "some other formulas that give the same result."
But this formula, to me, just seems so arbitrary and I would like to know how it's derived, and I would also like to know about these "other formulas." The linked post mentions "expressing its superfunction in a form of Newton series," and I have no idea what a Newton series is, and I don't even have a decent understanding of what a superfunction is.