Til the last breath

\[ \large \displaystyle \sum_{n=1}^{\infty} \dfrac{n(n+2)}{1\cdot2\cdot3+2\cdot3\cdot4+\cdots +n(n+1)(n+2)} \]

If the above expression is in the form \(\color{red}{ \dfrac{A}{B}}\) where \(\color{red}{A}\) and \(\color{red}{B}\) are co-prime positive integers, then let \(\color{darkgreen}{\phi}=\color{red}{A}\times \color{red}{B}\).

\[\large \color{purple}{\varphi}=\sqrt{\color{darkgreen}{\phi}\sqrt{\color{darkgreen}{\phi}^2\sqrt{\color{darkgreen}{\phi}^3\sqrt{\color{darkgreen}{\phi}^4\cdots}}}}\]

Find the value of \(\color{blue}{\left(\dfrac{\sqrt{\color{purple}{\varphi}}}{5}\right)!}\).

Notation:

\(!\) denotes the factorial notation. For example, \(10! = 1\times2\times3\times\cdots\times10 \).

Inspired by Rohit Udaiwal.

Try the Part-2 here