Exponential generating function with recurrent coefficients

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 :

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

Question : can we get a compact form for $F$?