As we see in the answers of this linked question which is counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $0\leq a_i\leq r_i$ for $i \in \{1,2,\ldots,n\}$ such that
$$a_1+a_2+a_3+\ldots+a_n=N$$
Mike Earnest answered like this:
$$ W = \sum_{S\subseteq \{1,2,\dots,n\}}(-1)^{|S|}\binom{N+n-1-\sum_{i\in S}(r_i+1)}{n-1} $$
But this is a little hard to calculate. Is there any approximation that be close to the answers of this statement?
Edit: Let me clear my question with some example. I test the result for $a_1+a_2+a_3=N$ such that
$$ 0 \leq a_1 \leq 1\text{,} \qquad 0 \leq a_2 \leq 3 \text{,} \qquad 0 \leq a_3 \leq 5$$
for $ 0\leq N \leq9$. This is how the result look like in system of coordinates with $N$ for $x$-axis and $W$ for $y$-axis
As we see by increasing $N$, $W$ begins from $1$ and hits a maximum value and then decreases to $1$ again.
But in unbounded case, it rises:
The reason that there is no maximum value for $0\leq N$ is that by increasing $N$ the bounds for $a_i$, which is $0 \leq a_i \leq N$, increase too, but in other case we have a fixed bound.
In this example by trial and error I found this polynomial approximation
$$ W = -\frac{2}{5}\left( N-\frac{9}{2}\right)^2 + 8 $$
It makes me wonder if can we find a polynomial approximation for $W(N,n,r_i)$ in any bounded case?