My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ .
It is practically the discrete convolution of these, which is a multiplication in the space of the Laplace/Fourier/$z$-transformations. Going in this direction I found complicated formulas whose de-transformation was already hopeless. (I could write them here on need.)
Was it a good direction? Maybe somebody knows the formula, or can show me a better way?