Consider some $\mathbb{N}_0$-valued random variable $X$. I am looking for an example where $$ \sum_{k=a}^\infty k^2 P(X=k) < \infty$$ for some $a \in \mathbb{N}_0$, i.e. the truncated second moment of $X$ exists, but $$ \sum_{k=0}^\infty k(k-1) P(X=k) = \infty,$$ i.e. the second factorial moment does not exist. I guess, a very similar, if not identical, question would be, when the truncated second moment exists, but the ordinary (not truncated) second moment does not. Any ideas?
Edit: As pointed out in a comment, there is no such example. (Should I delete this question then?)