I'm trying to find a neater expression for this sequence:
$$ \sum_{s \geq 0} e^{-\lambda}\frac{\lambda^{s}}{s!} sf(w)F(w)^{s-1} $$
where $f(w) = F'(w)$.
The following is my approach, but I think I did something horribly wrong with the sequence - I yield a factorial of $-1$ and don't know how to deal with that...
$$ \\ e^{-\lambda} f(w)\sum_{s \geq 0} \frac{\lambda^{s}}{(s-1)!} F(w)^{s-1} \\ e^{-\lambda} f(w)\sum_{s \geq -1} \frac{\lambda^{s+1}}{s!} F(w)^{s} \\ \frac{1}{\lambda}e^{-\lambda} f(w)\sum_{s \geq -1} \frac{\lambda^{s}}{s!} F(w)^{s} \\ \frac{1}{\lambda}e^{-\lambda} f(w)\left[ \frac{1}{-1!}F(w)^{-1} + \sum_{s \geq 0} \frac{\lambda^{s}}{s!} F(w)^{s} \right]\\ \frac{1}{\lambda}e^{-\lambda} f(w)\left[ \frac{1}{-1!}F(w)^{-1} + e^{\lambda F(w)}\right] \\ \frac{1}{\lambda}e^{-\lambda}f(w) \frac{1}{-1!}F(w)^{-1} + \int \frac{e^{-\lambda (1 - F(w))}}{\lambda}f(w)\\ $$
One may observe that the first term in your initial sum vanishes $$ \begin{align} \sum_{s \geq 0} e^{-\lambda}\frac{\lambda^{s}}{s!} sf(w)F(w)^{s-1}&=\sum_{s \geq 1} e^{-\lambda}\frac{\lambda^{s}}{s!} sf(w)F(w)^{s-1} \\\\&= f(w)\: \lambda e^{-\lambda}\sum_{s \geq 0}\frac{\lambda^{s}}{s!}F(w)^{s} \\\\&=f(w)\: \lambda e^{-\lambda}e^{\lambda F(w)} \end{align} $$ with a change of index in the last step.