For integer \( n \geq 1 \), define
\[
a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.
\]
Prove that \( a_n b_n \) is an integer.

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