The Lambert function has the following Maclaurin series:
$-W(-x)=\sum_{k=1}^\infty \frac{k^{k-1}}{k!} x^k$
(In some references, $-W(-x)$ is referred to as the "tree function", $T(x)$, as it is a generating function for rooted labeled trees.)
This series can be derived through Lagrangian inversion. In particular, the coefficient of $x^k$ in the power series of the tree function is given by the expression
$\frac1{k!}\left.\dfrac{\mathrm d^{k-1}}{\mathrm dt^{k-1}}\left(\frac{t}{t\exp(-t)}\right)^k\right\vert_{t=0}=\frac{k^{k-1}}{k!}$
The original equation can be re-expressed as
$\frac{2(n-1)n^{n-2}}{n!}=\sum_{k=1}^{n-1} \frac{k^{k-1}}{k!}\frac{(n-k)^{n-k-1}}{(n-k)!}$
The right hand side is the autoconvolution of the sequence $\dfrac{k^{k-1}}{k!}$; its ordinary generating function is thus the square of the generating function of $\dfrac{k^{k-1}}{k!}$, which is $(-W(-x))^2=W(-x)^2$.
To find an expression for the series coefficient of $W(-x)^2$, we consider first the related function $\left(\dfrac{W(-x)}{-x}\right)^2-1=\exp(-2\,W(-x))-1$. This function is the inverse of $\dfrac{\log\sqrt{1+x}}{\sqrt{1+x}}$. Applying Lagrangian inversion to $\dfrac{\log\sqrt{1+x}}{\sqrt{1+x}}$ and using the results obtained in this answer, we obtain the series
$\left(\frac{W(-x)}{-x}\right)^2-1=\sum_{k=1}^\infty \frac{2(k+2)^{k-1}}{k!}x^k$
which rearranges to
$W(-x)^2=\sum_{k=0}^\infty \frac{2(k+2)^{k-1}}{k!}x^{k+2}$
and can also be expressed as
$W(-x)^2=\sum_{k=1}^\infty \frac{2(k-1) k^{k-2}}{k!} x^k$
which proves the simpler expression for the autoconvolution.
The last series also appears as formula 11 of this paper by Corless, Jeffrey, and Knuth, as well as appearing in a disguised form as equation 5.60 in Concrete Mathematics.