For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, an expression of $F(x)$ as a function of $x$.