Let $f(x)=(1-x)^b (1+x)^{(n-b)}= \sum\_{i=0}^n a\_ix^i$, where $n$ is a positive
integer and $b$ 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 $b=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach
for this problem? 



Absolute sum of coefficient of (1-x)^b (1+x)^{(n-b)} 
Assume without loss of generality that $b\leq n/2$. Writing $f(x)=(1-x^2)^b(1+x)^{n-2b}$ shows that an upper bound is $2^{n-b}$, but this is very crude.


