Given a set of positive integers $N_1,N_2,...N_N$ and a function $f(x)$ where $f(x)=ℤ$ for all $x \in ℤ$. There exist solutions of the form $n_1,n_2,...n_N$ where $n_i \ge 1$ and $n_i \in ℤ$ subject to the constraint: $$\sum_{i=1}^k n_i < f\left(\sum_{i=1}^k N_i\right)$$
question: for a set $N_1,N_2,...N_N$; what is the total number of different solutions?
Whilst i have been able to manually calculate all comibnations for specific sets i have been unable to figure out a method which doesn't require brute force. I have been able to determine that the number of combinations is always less than: $$\prod_{i=1}^N f\left(\left(\sum_{k=1}^i N_k \right) - i\right)$$