I want to find the generating function of $f(n)=\sum _{k=0}^{n}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}(-1{)}^{n-k}{C}_{k}$, where ${C}_{k}$ is the k-th Catalan number. So, using the definition of an ordinary generating function:

$F(x)=\sum _{n\ge 0}{\textstyle (}\sum _{k=0}^{n}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}(-1{)}^{n-k}{C}_{k}{\textstyle )}{x}^{n}$

recalling that: $C(x)=\sum _{n\ge 0}{C}_{n}{x}^{n}=\frac{1-\sqrt{1-4x}}{2x}$

The first idea was to see F(x) as a product of two formal series and I have already seen a proof in this regard where they use the theorem of residues, yet I am looking for something less refined. Any idea?