Can this expression be further simplified : $ \sum_{k=0}^{n-1} { n -1 \choose k } (-2)^{k} (2n - k)! $? This is the coefficient of $x^{2n}$ in the formal power series expansion of $(1-2x)^{n-1} \times \sum_{ k \geq 0} k! x^k$.
Motivation: I came across this when trying to solve a problem using inclusion-exclusion principle, I am not mentioning the original problem because I am interested in this sum as an independent problem.