I was going through the cycle index topic and I came across the cycle index for various types of groups. It is given that the cycle index for a cyclic group is $$P_{C_n} (t_1 , \ldots ,t_n)= \frac{1}{n} \sum_{d\mid n} \phi (d) t_d^{\frac{n}{d}},$$ Similarly the cycle index for a dihedral group with $2n$ elements is given as $$P_{D_n}(x_1, x_2, \ldots x_n) = \frac{1}{2} P_{C_n} (x_1, x_2, \ldots x_n) + \left\{ \begin{array}{l} \frac{1}{2} \left( x_1x_2^{(n-1)/2} \right) \textrm{ when $n$ is odd } \\ \frac{1}{4} \left( x_2^{n/2} +x_1^2x_2^{(n-2)/2} \right) \textrm{ when $n$ is even } \end{array} \right. $$
Is there any way to derive those?