Chromatic Polynomial
You're using a different numbering than Wikipedia; your $W_n$ is Wikipedia's $W_{n+1}$. Thus you need to substitute $n+1$ for $n$, yielding the chromatic polynomial
$$x((x-2)^n-(-1)^{n+1}(x-2))\;.$$
To find this polynomial, note that you need one colour for the centre and the remaining $x-1$ colours for the remaining vertices, which form a cycle $C_n$. The number of colourings of $C_n$ with $r$ colours is calculated in this answer, and substituting $x-1$ for $r$ and multiplying by $x$ for the number of choices for the colour of the centre yields the above polynomial.