Combinatorial Identities involving roots of unity

Question

I'm looking for combinatorial identities involving roots of unity/complex numbers and binomial coefficients. In Gould's Combinatorial Identities, I have come across $$\sum_{j\geq0} \binom{n}{rj} = \frac{1}{r} \sum_{j=0}^{r-1} (1 + w^j)^n$$ and $$\sum_{k\geq 0}\frac{1}{\binom{kr}{r}}=\sum_{k=1}^{r-1}-\omega^k(1-\omega^k)^{r-1}\log\frac{1-\omega^k}{-\omega^k}$$ where $\omega$ is a primitive $r$-th root of unity. I am looking for more identities of this flavor.