I'm wondering if there is anything known about the asymptotic behavior of the sequence $a_n=W(n,1)$. Here, $W(n,x)$ represents the $n^{th}$ branch of the Lambert W function, where by definition $x = W(n,x) \exp(W(n,x))$ for all $n$. I've made a plot of the of the first few $n$. The plot shows $W(n,1)$ in the complex plane.
A plot of $a_n$.
Clearly the points lie along some nice curves - what I would ideally like is an expression for these curves, at least in the large $n$ limit. Does anybody know of such an expression, or a reference where such a thing is derived?