# Learning how combinatorial expressions relate to integration with complex numbers

I was playing around with some combinatorial expressions and tried to simplify the expression $$\binom{k^2+k}{k^2-k}$$. I got stuck at the point $$\prod_{m=1}^{m=2k} \frac{k^2-k+m}{m}$$ so I plugged the expression $$\binom{k^2+k}{k^2-k}$$ into Wolfram Alpha and got the following equivalent integral representations I would like to know what resources I should use to learn about how this simplification occurs. Currently I am knowledgeable about single-variable Calculus, multi-variable Calculus, and differential equations, the latter two to a much lesser extent.

The right-hand side is $$\sum_{j=0}^{k(k+1)}\frac{1}{2\pi}\int_{-\pi}^{\pi}\binom{k(k+1)}{j}\exp i(j+k-k^2)tdt$$. But any integer $$l$$ satisfies $$\frac{1}{2\pi}\int_{-\pi}^\pi\exp iltdt=\delta_{l0},$$so the only surviving term has $$j=k^2-k$$.