Let's say we have a multivariate polynomial generating function $$f(x_1,...,x_m) = \sum_{0\le n_1,...,n_m\le k}a_{n_1,..,n_m} x_1^{n_1}\cdot ...\cdot x_m^{n_m}$$
I now want to extract a partial sum of its coefficients, namely $$ \sum_{c_1\,\le n_1\,\le k\\...\\c_m\le n_m\le k}a_{n_1,..,n_m} $$
One possible way would be to use the process of differentiating to extract a single coefficient, and then repeat this for all possible tuples $(n_1,...,n_m)$, i.e.:
$$\sum_{i_1=c_1}^k ...\sum_{i_m=c_m}^k \left[\left(\frac d{dx_1}\right)^{i_1}...\left(\frac d{dx_m}\right)^{i_m}\frac{f(x_1,...,x_m)}{i_1!...i_m!}\right]_{x=0}$$
Are there any alternatives leading to nicer formulas?