Sum[(-n)^(n-1)/n! z^n,{n,\[Inf]}] == ProductLog[z]

[John Conway]

On Sun, 9 Aug 1998, Wouter Meeussen wrote:
> 
> I take it that Lambert_W is the same as what Mathematica calls "ProductLog".
> (quote:  ProductLog[z] gives the principal solution for w in  z = w  e^w  )

   I thought at first that this was what was traditionally called the
"Zech's logarithm", Z(x), defined by  Z(log(x)) = log(x+1), but it doesn't
seem to be.  May I ask about series expansions for that?

> I most certainly find its Series development Weird ! it's (-n)^(n-1)
> *************************************************************************
> this means that
> 
> Sum[(-n)^(n-1)/n! z^n,{n,\[Infinity]}] == ProductLog[z]

    I'm fairly sure that this is one of the standard examples for the
theorems of Burmann and Lagrange that show how to express one function
as a power series in terms of another.

    John Conway