<s>[INST] Coefficient of $x^{n-2}$ in $(x-1)(x-2)(x-3)\dotsm(x-n)$ Question

Find the coefficient of $x^{n-2}$  in the expression $$(x-1)(x-2)(x-3)\dots(x-n)~~.$$

My approach

The coefficient of $x^n$ is $1$. The coefficient of  $x^{n-1}$ is $- \frac{n(n+1)}{2}$

But I cannot proceed from here.
I would appreciate any help.
 [/INST] Taking the long way:
\begin{align}
f_{n}(x) &= (x-1)(x-2)\cdots(x-n) \\
f_{1}(x) &= x-1 \\
f_{2}(x) &= x^2 - 3x + 2 \\
f_{3}(x) &= x^3 - 6x^2 + 11x - 6\\
f_{4}(x) &= x^4 - 10x^3 + 35x^2 = 50x + 24 \\
f_{5}(x) &= x^5 - 15x^4 + 85x^3 - 215x^2 + 274x - 120.
\end{align}
From here it is determined that $[x^n] \, f_{n}(x) = 1$, $[x^{n-1}] \, f_{n}(x) = -\binom{n+1}{2} = s(n+1,n)$ and $[x^{n-2}] \, f_{n}(x) \in \{ 2, 11, 35, 85, 175, \cdots \}$. This pattern follows the (signed) Stirling numbers of the first kind, $s(n+2,n)$ or 
$$[x^{n-2}] \, f_{n}(x) = \frac{(n-1) (n) (n+1) (3 n + 2)}{4!} = s(n+1, n-1).$$ 
One may compare the values obtained to A000914.
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