$\require{begingroup} \begingroup$
$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
This is a follow up question to the recently asked one.
Is there a closed form for the sum
\begin{align} S&=\sum_{n=1}^\infty \frac{\Gamma(n+2,n+1)}{\Gamma(n+2)\,n^3\,(1+\tfrac1n)^{n+1}} \tag{1a}\label{1a} \\ &= 1-\frac1\e\,\left(2+\int_0^1 \frac{\Wp(-\tfrac t\e)}{\Wm(-\tfrac t\e)}\, dt\right) \approx 0.20861152792812846 \tag{1b}\label{1b} , \end{align} where $\Wp,\Wm$ are the two real branches of the Lambert W function?
Not sure, if this is of any use, but the numeric value of the sum is surprisingly close to $\frac\Omega\e$, where $\Omega=\W(1)$,
\begin{align} \left|S-\frac\Omega\e\right|&<3\cdot10^{-5} \tag{2}\label{2} . \end{align}
Related sequence of fractions \begin{align} A_n&=\exp(n+1)\cdot \frac{\Gamma(n+2,n+1)}{\Gamma(n+2)\,n^3\,(1+\tfrac1n)^{n+1}} \tag{3}\label{3} \end{align}
is: \begin{align} \left[ \frac54, \frac {13}{27}, \frac {103}{256}, \frac {4388}{9375}, \frac {30575}{46656}, \frac {850914}{823543}, \frac {1335923603}{754974720}, \frac {43671523328}{13559717115}, \frac {429970430763}{70000000000}, \dots \right] \tag{4}\label{4} , \end{align}
The numerators and denominators of \eqref{4}:
\begin{align} &\left[ 5, 13, 103, 4388, 30575, 850914, 1335923603, 43671523328, 429970430763, \dots \right] \tag{5}\label{5} ,\\ &\left[ 4, 27, 256, 9375, 46656, 823543, 754974720, 13559717115, 70000000000, \dots \right] \tag{6}\label{6} . \end{align}
$\endgroup$
This is not an answer but it is too long for a comment.
For sure, it is really nice to see $\Omega$ and $e$ appearing so simply and I like it, be sure.
However, at this level of accuracy, you could find many coincidences. For example $$S=\frac{53+1367 e-391 e^2}{3 \left(-236+237 e+135 e^2\right)}$$ is correct for almost $20$ significant figures.
In the same way, $$\int_0^1 \frac{W_0(-\frac te)}{W_{-1}(-\frac te)}\, dt\sim \frac{7188}{23885+2 \sqrt{169014673} \cos \left(\frac{1}{3} \cos ^{-1}\left(\frac{617823390329}{169014673 \sqrt{169014673}}\right)\right)}$$is correct for almost $20$ significant figures. This is the reciprocal of the largets root of $2396 x^3-23885 x^2+55854 x-17749=0$.