Explicit expressions for the $c_0,\ldots,c_{p-2}\in\Bbb{Z}$ satisfying $(1-\zeta_p)^{p-1}=p\sum_{k=0}^{p-2}c_k\zeta_p^k$.

Question

It is well known that in the cyclotomic ring $\Bbb{Z}[\zeta_p]$ one has the equality of ideals $$(1-\zeta_p)^{p-1}=(p).$$ I'm struggling to find 'manageable' expressions for the associated units $$\frac{(1-\zeta_p)^{p-1}}{p}\qquad\text{ and }\qquad\frac{p}{(1-\zeta_p)^{p-1}},$$ on the power basis $\zeta_p^0,\ldots,\zeta_p^{p-2}$ of $\Bbb{Z}[\zeta_p]$. One approach I tried uses that $$(1-\zeta_p)^{p-1}=\prod_{n=1}^{p-1}(1-\zeta_p^n)\frac{1-\zeta_p^n}{1-\zeta_p}=\Phi_p(1)\prod_{n=1}^{p-1}\frac{1-\zeta_p^n}{1-\zeta_p}=p\cdot\prod_{n=1}^{p-1}\sum_{m=0}^{n-1}\zeta_p^m,$$ which gives the (to me) unmanageable expression $$\frac{(1-\zeta_p)^{p-1}}{p}=\prod_{n=1}^{p-1}\sum_{m=0}^{n-1}\zeta_p^m.$$ I've written this out explicitly for $p\leq7$, but I don't see a pattern yet. Are there nice expressions for the associated units above?

EDIT: I like to clarify that I'm looking for explicit expressions for the coefficients of these units on the standard power basis for $\Bbb{Z}[\zeta_p]$. And although the answer by sharding4 below does exactly that, I was hoping for expressions without denominators. That is, expressions from which it is immediately clear that this is an element of $\Bbb{Z}[\zeta_p]$.

Answer 1

A fairly "manageable" expression in terms of the power basis can be obtained simply by applying the binomial theorem to $(1-\zeta_p)^{p-1}$ and using the fact that $\zeta_p^{p-1}=-\zeta_p^{p-2}-\zeta_p^{p-3}-\cdots - \zeta^2-\zeta - 1$. Then $$ \frac{(1-\zeta_p)^{p-1}}{p}=\sum_{k=1}^{p-2}\frac{(-1)^k\binom{p-1}{k}-1}{p}\zeta_p^{k} $$