Let $S_n=1 \cdot(n-1)+2 \cdot(n-2)+3 \cdot(n-3)+\ldots \cdot+(n-1) \cdot n \geq 4$.
The sum $\sum_{n=4}^{\infty}\left(\frac{2 S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to :
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