Infinite Factorial Summation with Sum of the powers of 2

Calculus Level 4

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

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 \(A\times B \times C\).