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.



Can this function be interpolated with a small power series
Take an entire function $f$ such that $f(0)=1$ and $f(j) = 0$ for all nonzero integers: an example is $f(z) = \sin(\pi z)/(\pi z) $ for $z \ne 0$, $1$ for $z=0$.  

The Maclaurin series of $f$ satisfies $\sum\_{i} |a\_i| < \infty$.


