Does there exist a (sequence of) power series $\sum\_{i\geq 0} a\_{n,i} x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum\_{i\geq 0} \vert a\_{n,i}\vert n^i=O(n^p)$ for some real $p>0$?



Small power series "approximating" a Dirac
Such functions do not exist. Let us call them $f\_n$. Your conditions imply that
$f\_n(0)=1, \; f(k)=0$ for $ 1\leq k\leq n-1$ and $M(n,f\_n)\leq Cn^p$, where $M(r,f):=\max\{|f(z)|:|z|=r\}$. Now [Jensen's Formula](https://en.wikipedia.org/wiki/Jensen%27s_formula) says that
$$\sum\_{k=1}^{n-1}\log\frac{n}{k}\leq \frac{1}{2\pi}\int\_0^{2\pi}\log|f\_n(ne^{i\theta})|d\theta.$$
The RHS is at most $p\log n+\log C$, while the LHS is
$$n\log n-\log n!\sim n$$
by Stirling's formula. So we obtain a contradiction.


