Suppose I have a sequence of real numbers $a_n$ and a real analytic function $f$ such that at a point $x$ (maybe 0), we have $f^{(n)} (x) = a_n$, for all $n$. Can we say anything interesting about this function in terms of $a_n$?
More specifically, are there any specific examples of sequences that might arise in other contexts where the corresponding function has interesting or surprising properties? For example, I first thought of this question when wondering what the properties of a function whose derivatives at a point form the Fibonacci sequence might be.