I'm trying to prove that the series below converges to 1 and I noticed it looked strikingly similar to a probability distribution I once saw. My question is twofold:

- Can anyone identify the distribution? I can't seem to, for the life of me, remember. I'm 90% sure this is a probability distribution (or some form of one) but I may be wrong.
- Does anyone have any hints for proving this convergence? I don't necessarily want an answer, just some pointers. I don't think this requires any complex mathematics beyond an early graduate course in analysis.

$$ \sum_{j=1}^{\infty}{\frac{e^{-j}j^{j-1}}{j!}}= 1 $$

Thank you!