Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$?

I feel the answer might be no but I'm not sure how to prove it. Perhaps this question isn't appropriate for this site, feel free to remove it if so.