Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that

$$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$

So for any positive integer $n$, we can write

$$(1 - \zeta_p)^n = \sum_{i=0}^{p-2} c_{i,n} \zeta_p^i$$

for some $c_{i,n} \in \mathbb{Z}$. There is clearly a recursive formula for these coefficients.

Question: Is there a known closed form for the coefficients, $c_{i,n}$, in terms of $p$, $n$, and $i$?

I have found a solution for $p=3$, but $p=5$ is already quite difficult. The pattern seems to repeat mod $p(p-1)$. Additionally, while I personally am not thinking of these $p$th roots of unity as lying in $\mathbb{C}$, the question could also be viewed in that way by replacing $\zeta_p$ with $e^{2\pi i/p}$.

Any help would be much appreciated!