From wolfram mathematica I got this result.
$\sum\limits_{i=0}^{k}\frac{k!x^i}{i!(k-i)!^2}$
This is $\frac{e^{i\pi k}x^k U(-k,1,-1/x)}{k!}$
Where U(a,b,z) is the confluent hypergeometric function of the second kind. I tried to plot it in Mathematica but it doesn't show anything. How can I plot this for very big k with respect to x? The goal is to plot $n^k\frac{e^{i\pi k}x^k U(-k,1,-1/x)}{k!}$ where n is fix but big as you want and k goes to infinity to see when that goes to zero