In view of Iosif Pinelis's [question](https://mathoverflow.net/questions/377012/an-identity-for-the-lambert-w-function) about $\sum_{k\in\Bbb Z}1/(1+W_k(x))$, I played a little with
$S(x,j)=\sum_{k\in\Bbb Z}1/W_k(x)^j$. It seems that if $x$ is a rational number with small denominator, so is $S(x,j)$. My quick and dirty implementation may be wrong, but I seem to find
for instance:
$$\begin{array}{c|ccccc}
& j = 1 & j = 2 & j = 3 & j = 4 & j = 5 \\ \hline
x = -0.1 & -19/2 & 80 & -715 & 19180/3 & -228675/4 \\
x = -0.2 & -9/2 & 15 & -115/2 & 665/3 & -20525/24 \\
x = -0.3 & -17/6 & 40/9 & -235/27 & 1420/81 & -34675/972
\end{array}$$
Similarly for other values. Is this true? Is it well known? What is the proof?

Added: for $x=-\exp(-1)$ we seem to have $S(x,j)=(-1)^j2$.

OK, I think I solved it, but I think it is interesting nonetheless:
these are the coefficients of $T^j$ of $(1/x)(1+T)/(\exp(-T)-T/x)$ except for $j=1$. This must be Lagrange inversion.