Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \ 17582819/12,\dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

Is this correct? I can't find a nice formula for the coefficients - can anyone help?