$\frac {1^7}{1!} + \frac {1^7 + 2^7}{2!} + \frac {1^7 + 2^7 + 3^7}{3!} + \frac {1^7 + 2^7 + 3^7 + 4^7}{4!} + \ldots$

If the series above equals to $W$, what is the value of $\frac {24}{e} \times W$?

Note: $e = \displaystyle \lim_{n \to \infty} \left (1 + \frac 1 n \right )^n$