Generating function of a parametrized binomial coefficient Let be $m$ an integer and $A_p(m) = \binom{mp}{p}$.
I'd like to know more about $B_m(z) = \sum_{p \geq 0} A_p(m) z^p$.
At least, I'd love to be able to compute $B_m\left(\dfrac{1}{q}\right)$ for some $q$ integers.
What I tried:


*

*Look at Fuss-Catalan numbers and its generating function to derive a relation with $B_m$

*But as there is no closed form of the generating function, I cannot derive an interesting enough relation here.


Intuitively, I could try to interpret $A_p(m)$ as something combinatorial and look for a recurrence relation to explicit $B_m$, but I have no idea.

This is not an answer.
For $m=1,2,3$ there are closed forms for $B_m(z)$. 
For $m\geq 4$ come again hypergeometric functions with interesting patterns
$$B_m(z)=\,
   _{m-1}F_{m-2}\left(\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}m; \frac{1}{m-1},\frac{2}{m-1},\cdots,\frac{m-2}{m-1};\frac{m^m}{(m-1)^{m-1}}z\right)$$
