On Non homogeneous linear recurrence relation and power series associated with it:

Question

Consider the following Non homogeneous recurrence relation :

$$T(2n) = nc_{n} + \sum_{k=1}^{n-1} c_k T(2(n-k))$$

Now consider the following power series :

$$F(x) = \sum_{n=1}^\infty T(2n) x^n$$

Question : Other possible representation of $F(x)$ with explanation?