calculating limit by second moment argument

Question

Let $0<p<1$ be fixed.

Let $X_n$, $n=1,2,...$, be a sequence of random variables with non-negative integer values.

Suppose their expectations satisfy $E(X_n)=\frac{n(n-1)}{2} p^{n-2}$.

Question : How can we use a standard second moment argument to prove $\lim_{n\to\infty} \text{Prob}\{X_n=0\}=0$?

Reference:

Linial, Nathan; Meshulam, Roy, Homological connectivity of random 2-complexes, Combinatorica 26, No. 4, 475-487 (2006). ZBL1121.55013.