arithmetic circuits for polynomials (updated) I am concerned about the  following question, consider
$P_n(x)= \sum_{i=0}^n \frac{x^n}{n!}$

Is there a straight line program (or arithmetic circuits) of polynomial size (wrt $n$) for the  polynomial $P_{2^n}(x)$? The difficulty seems to lie in the factorial, as for example the polynomial $x^{2^n}$ has a polynomial size arithmetic circuit (by repeated squaring). 

Sorry in my previous post I did not make the question clear, as it would become a very simple question.
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.