Properties of Inverse Lambert W function

Question

Starting with the two-branched Lambert W function (from Wikipedia):

Suppose we just flip it like this:

Is there a single power series for this $y=W^{-1}(x)$?

Answer 1

Lambert W function, $y=W(x)$ is a solution for $y \mathrm{e}^y = x$. Hence $W^{-1}(y) = y \mathrm{e}^y$.