Say I have a Diophantine equation of the form $a_1 x_1 + a_2 x_2 + ... + a_m x_m = n$ such that the $a_is$ are all co-prime to each other. And I also have a function say $f$ which depends only on the $x_i's$ (and will be evaluated on solutions of the equations)

Is there a general method or simple examples of summing over the values of $f$ evaluated on the non-negative integral solutions of the equation?

Is there a way to count the number of non-negative integral solutions of such Diophantine equations? (...I am aware that it is trivially doable in some special cases like when all the $a_is$ are equal to $1$ or when $a_i = i$ and $m=n$...)