Binomial Coefficients, Roots of Unity and Powers of Prime Numbers

Let $$t\in {\mathbb {N}}_+$$ be given. In this article, we are interested in characterizing those $$d\in such that the congruence $$\begin{aligned}\frac{1}{t}\sum _{s=0}^{t-1}{n+d\zeta _t^s\atopwithdelims ()d-1}\equiv {n\atopwithdelims ()d-1}\pmod {d}\end{aligned}$$ is true for each $$n\in {Z}}$$ . particular, assuming d has a prime divisor greater than t, show above holds if and only $$d=p^r$$ , where p number t $$r\in \{1,\ldots ,t\}$$