We show that Lagrange inversion can be used to obtain closed-form expressions for a number of series expansions of the Lambert $W$ function. Equivalently, we obtain expressions for the $n$th derivative. Various integer sequences related to the series expansions now can be expressed in closed form.
['David J. Jeffrey', 'German A. Kalugin', 'N. Murdoch']
Lagrange Inversion and Lambert W