Assume I have coins of four different denominations $d_i$, say $1$, $2$, $5$ and $10$. One problem that could be asked is how many different ways there are to get to $20$ by adding up exactly $3$ of these coins, where each one may be taken an unlimited amount. In that case, the solution would be to find the coefficient of $x^{20}$ in $(x^1 + x^2 + x^5 + x^{10})^3$, which in this case becomes $3$.
I could also ask how many ways there are to take $3$ coins and get less than $20$, which is just the sum of the coefficients of all powers of $x$ less than $20$, which in this particular example becomes $51$.
Now, let's say I take each coin through a function and change each value $d_i$ to some $f(d_i)$. My question is the following: is there a general way to count how many of the above $51$ combinations of coins would still sum to less than $20$ when the combinations are done with $f(d_i)$ replacing each $d_i$ in the sum?
Any help is appreciated!