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.

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.