$\large S= 1 + \dfrac{1+2}{2!} + \dfrac{1+2+2^2}{3!} + \dfrac{1+2+2^2+2^3}{4!} + \cdots$

Given that $S = Ae^2 + Be + C$, where $A,B$ and $C$ are integers, and $\displaystyle e= \lim_{x\to 0} (1+x)^{1/x} \approx 2.718$, find the value of $A\times B \times C$.