Expansion of Gaussian function in a series of exponential functions

Trying to solve a complicated integral, I got interested as a side question in the following problem. Suppose that $f(x)=\exp(-x^2/2)$ is a standard Gaussian function. Is it ever possible to represent $f(x)$ pointwise as $$f(x)=\sum_{k=-\infty}^\infty c_k (e^{-x})^k\ ,$$ i.e. as an infinite superposition of real exponentials? I have gut feelings pointing in opposite directions. I ultimately came to the conclusion that it should probably be 'possible', but for coefficients $c_k$ which are not easy to express in simple form. But it could also be that there is some obvious obstacle that at present I fail to see. Many thanks folks for your help (I am particularly interested in the possibility of writing the $c_k$'s in 'reasonable' form).