Question:

Sequence $\{a_n\}$ with $a_1=a_2=1$, $a_3=3$, $a_n=(n+3)a_{n-1}-na_{n-2}-(n-2)a_{n-3}$ for $n\ge 4$. Then $\dfrac{a_{2023}-a_{2020}}{2021!}=$