$\frac{2}{1!}(\log_e 2) + \frac{2^2}{2!}(\log_e 2)^2 + \frac{2^3}{3!}(\log_e 2)^3 + \dots \infty = $

The value of the infinite series $\frac{1^{2}+2^{2}}{3 !} + \frac{1^{2}+2^{2}+3^{2}}{4 !} + \frac{1^{2}+2^{2}+3^{2}+4^{2}}{5 !} + \dots$ is:

In the expansion of $\frac{e^{5x} + e^x}{e^{3x}}$,the coefficient of $x^4$ is

The sum of the infinite series $1 + 2 + \frac{1}{2!} + \frac{2}{3!} + \frac{1}{4!} + \frac{2}{5!} + \dots$ is

Let $\sum_{n=0}^{\infty} \frac{n^3((2n)!) + (2n-1)(n!)}{(n!)((2n)!)} = ae + \frac{b}{e} + c$,where $a, b, c \in \mathbb{Z}$ and $e = \sum_{n=0}^{\infty} \frac{1}{n!}$. Then $a^2 - b + c$ is equal to $................$.

${\left[ {1 + \frac{1}{{2!}} + \frac{1}{{4!}} + \dots \infty } \right]^2} - {\left[ {1 + \frac{1}{{3!}} + \frac{1}{{5!}} + \dots \infty } \right]^2} = $