Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$