I'm looking for an asymptotic upper bound for the function f(x), which is recursively defined as follows: $$f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$$

with $f(1)=1$. I am pretty sure that $f(x) \in x^{O(log x)}$, and this is the best one can get, but I don't know how to prove it analytically. Any ideas?