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In this note, we investigate some properties of the integer sequence ofgeneral term $a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)!$ ($\forall n \geq 1$)to derive a new identity of the Genocchi numbers $G_n$ ($n \in \mathbb{N}$),which immediately shows that $G_n \in \mathbb{Z}$ for any $n \in \mathbb{N}$.In another direction, we obtain nontrivial lower bounds for the $2$-adicvaluations of the rational numbers $\sum_{k = 1}^{n} \frac{2^k}{k}$.

The integrality of the Genocchi numbers obtained through a new identity and other results