Let $f(x)=(1-x)^e (1+x)^{(n-e)}= \sum_{i=0}^n a_ix^i$, where $n$ is a positive integer and $e$ is a non-negative integer less than $n$. I want to find an upper bound on $\sum_{i=0}^n |a_i|$ other than the trivial upper bound $2^n$. Also for $e=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach for this problem?