I have the following recurrence:

Let's say that I have 1 array f and 1 function g. A function call to g takes O(1) time. I just want to compute the values in the f array using the recurrence given below:

f[n] = summation(f[n — i]*g(i, n)) for i in [1, 2, 3, 4, ..., n]

Base Case: f(0) = 1.

If we use the above recurrence, it will take O(n^2) time. Any way to reduce it to O(n*log(n)) ?

Thanks in advance.