I am looking for a testfunction $\phi\in\mathcal{D}(\mathbb{R})$ (that is a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ that is in $C_c^{\infty}(\mathbb{R})$) which $n$-th derivatives take the same (non zero) value at a given point $x$ ($x=0$ for example).
So I want to find a function $\phi$ such that $D^n\phi(0)=a\neq0$ for all $n\in\mathbb{N}$.
All I can think of are functions of the type $e^{-\frac{1}{1-x^2}}\chi_{[-1,1]}$ but then the derivative changes and I am not even sure if such a function even exists.
Another thing I wondered about was if there is a function $\phi$ such that $D^n\phi(0)$ is not summable so that we have $\sum_n^{\infty}D^n\phi(0)=\infty$. In particular this should then disprove that the functional $\Lambda_\phi=\sum_n^{\infty}D^n\phi(0)$ is a disribution.
Let $f(x)=ae^x$ so that $f^{(n)}(0)=a$ for all $n\in \Bbb{N}_0$, and let $\psi$ be a smooth compactly supported function which is identically equal to $1$ in a neighborhood of the origin. Then, $\phi(x)=f(x)\psi(x)$ works.