Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$. 

One needs to evaluate $F(X)$ at $O(2n)$ points to interpolate and get all the coefficients of $F(X)$. 

However say I need **only the coefficient of $X^{2n}$**, do I still have to evaluate at $O(2n)$ points? 

Will having coefficient of $X^{t}$ same as coefficient of $X^{2n-t}$ help in reducing the number of points from $O(2n)$? 

Is this a well studied problem that has some good references - that is interpolating for only one or few coefficients?