Discrete distribution where mgf exists only at zero but all moments are finite

Question

Does such a distribution exist and if it does, what does it look like? For the continuous case, there is the log-normal distribution, so my gut says there must be an analogous discrete distribution but I'm having a hard time constructing it.

Answer 1

Discrete example, based on your continuous one: $P(X=n)\propto\frac{1}{|n|}\exp-\frac{\ln^2 |n|}{2}$ for integers $n\ne 0$, the proportionality constant determined by unitarity. Proof of validity is an easy exercise with the integral test for convergence. I've made the distribution symmetric to ensure even negative-real-part arguments for the mgf don't let it converge.