Concerning the other coefficients $c_k$, Branan [1] showed that $(k+1)c_{k+1}=(n-k)\bar c_{n-k}$ is a necessary condition for univalence in $|z|<1$ if $c_1=1$ and $c_n=1/n$.
For univalent polynomials with real coefficients and such that $c_{n}=1/n$, Suffridge [2] has proved that the coefficients $c_{k}$ satisfy the inequalities $$|c_{k}|\leq |C_{k,1}|, $$ and for univalent polynomials with real coefficients and such that $c_{n}=-1/n$, $$ |c_{k}|\leq\begin{cases} |C_{k,1}|\quad(n\text{ even})\\ |C_{k,2}|\quad(n\text{ odd}), \end{cases} $$ where $$C_{k,j}=\frac{n-k+1}{n}\frac{\sin jk\pi/(n+1)}{\sin j\pi/(n+1)},\qquad k=2,\ldots,n,\quad j=1,\ldots,n.$$ For each $n=1,2,\ldots$, and $1\leq j\leq n$, the polynomials $$P(z)=\sum_{k=1}^n C_{k,j}z^k$$ are univalent in $|z|<1$, from which follows that the above inequalities are sharp.
[1] D. A. Brannan, Coefficient regions for univalent polynomials of small degree, Mathematika, 14 (1967), 165-169.
[2] T. J. Suffridge, On univalent polynomials. J. London Math. Soc. 44 (1969), 496-504.