arithmetic circuits for polynomials (updated)
I assume that you mean $P_n(x)= \sum_{i=0}^n \frac{x^i}{i!}$.

Peter Bürgisser shows that if this problem is hard, then computing the permanent
is hard (http://link.springer.com/article/10.1007/s00037-009-0260-x).
To my best knowledge, your problem is open.