Consider the following question:
What's the coefficient of $z^4$ where
$\Pi_{k=1}^{7}(z-e^{ik\pi}), \quad z\in{\bf C}$
I am wondering whether there is some "trick" for solving the problem above quickly, instead of simply expanding the product or using the Taylor expansion. More generally,
What's the coefficient of $z^m$ where
$\Pi_{k=1}^{n}(z-e^{ik\pi}), \quad z\in{\bf C}$
What's the topic to which the above question is related in complex analysis and where such formula might appear?