Lets define discrete $ f_N(i) = 1,\space i = 1...N $
I need to find $ G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m} $
For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , which is palyndromic and have $\binom{i}{2} $ as first 5 values.
So it seems to be common rule, every $G_N^m$ is palyndromic, have $mN - m + 1$ values total and have $\binom {i} {m-1}$ as first $ N-1$ values.
But i wonder if there general formula for $G_N^m(i)$ ?
Additional information about discrete convolution: values of $G_N^m$ are coefficients of $(\frac{x^N - 1}{x -1})^m$ polynomial