For example, how many ways can we write 20 as a sum of 2, 9, and 11?

To make things easier, let's suppose that different arrangements of the same combination are counted separately. Therefore, something like 9 9 2 and 9 2 9 will be counted as 2 separate ways.

Is there a general formula for cases like these?

The best I could come up with is a recursive formula.

For a number n and a set (a1, a2, a3, .....), let's suppose that F(n) is a function that can tell us the number of ways to write n as a sum of numbers from the set (a1, a2, a3, ......).

F(n) = F(n-a1) F(n-a2) F(n-a3) ......

But I don't know how I can do this without regression. Heck, I don't even know if my idea is correct.