Convergence rate of $(1+x/n+\dots)^n$

It is a well known fact that

$$\lim_n (1+x/n+O(n^{-3/2}))^n=e^x$$

For example, this is a key step in the standard proof of the central limit theorem.

What can we say about the rate of convergence of this approximation?

More precisely, suppose we expand out the first term of the remainder:

$$(1+x/n+y/n^{3/2}+O(1/n^2))^n$$

Is it correct to say that this is equal to $e^x+C/\sqrt{n}+O(n^{-3/2})$? And if so, is there an explicit expression for $C$ in terms of x and y?