Bound on derivatives of compactly supported function

Let $f$ be a smooth function of compact support on $\mathbb R$.Let$$a_n=\sup_{x\in\mathbb R}|f^{(n)}(x)|.$$I would be interested in upper bounds of the growth of this sequence. In particular: does there exist a constant $C>0$ such that$$a_n\le C^n\ n!$$where $n!=1\cdot 2\cdots n$.