Over the last few weeks I have become obsessed with expressions like $$ \frac{e+4 e^{2}+e^{3}}{(1-e)^{4}}, $$ $$ \frac{e+26 e^{2}+66 e^{3}+26 e^{4}+e^{5}}{(1-e)^{6}}, $$ or $$ \frac{e+120 e^{2}+1191 e^{3}+2416 e^{4}+1191 e^{5}+120 e^{6}+e^{7}}{(1-e)^{8}}. $$
These expressions sparked interest in me because they approximate $3!,5!$ and $7!$, respectively and are part of a larger family of decent approximations (these approximations are quite good till 16!).
One reason why this is true is because the relation between this expressions and polylogarithm, and the later with the gamma function, but I would really like to know if there is another way to justify why they approach factorials.
I have delved on analytic combinatorics, and $q$-analogs that talk about evaluating at roots of unity, poles or saddle points, but none of them seem totally appropriate, Although they study the idea of evaluating a generating function in transcendental or complex numbers, none of them seem to relate directly to this.
PD: a Dual to this identities is the evaluation at $e^{-1}$, $$\frac{e^{-1}+11 e^{-2}+11 e^{-3} +e^{-4 }}{\left(1-e^{-1}\right)^{5}}$$ these are the same but with a correcting minus sign, so with $e^{-1}$ we get $4!$ while with $e$ we get $-4!$. I add these as they make more sense when we convert back to the classic polylogarithms expression $$5! \approx \operatorname{Li}_{-5}(z) = \sum_{k=0}^ \infty k^{5}z^k$$ with $z=\frac{1}{e}$ that needs $|z|<1$, but I guess the $e$ expressions coincide with the analytic continuation of these series
Also Something that made me stick with this subject was that for example $$ \frac{z+26 z^{2}+66 z^{3}+26 z^{4}+z^{5}}{(1-z)^{6}} =\frac{1}{z-1}+\frac{31}{(z-1)^{2}}+\frac{180}{(z-1)^{3}}+\frac{390}{(z-1)^{4}}+\frac{360}{(z-1)^{5}}+\frac{120}{(z-1)^{6}} $$ where something notable is that the last coefficient is exactly 5!, this is because eulerian numbers sum to factorials, but this also seemed crazily connected with the Cauchy residue theorem, only that this one was about $(z-1)^{-n}$ and not about $(z-1)^{-1}$
EDIT: Someone edited out an oeis entry linking this to the eulerian polynomials, so just so you know I'm aware what they are, also I will add a comment that I think it's crucial to finding the connection
"This seems to be connected to this combinatorial problem math.stackexchange.com/questions/257890/simon-newcombs-problem stating the formula $$\sum_{d=0}^{\infty} \frac{A_{d}(t) x^{d}}{(1-t)^{d+1} d !}=\frac{1}{1-t e^{x}}$$ the wikipedia page has the case $x=1$ although it doesn't seem to provide any source, while Jair answers [Comment] is that formula with $t=1/e$. This seems promising as, that formula is almost a composition of classical analytical combinatorics constructions, and the wikipedia case has a $1/e$ as pole. "
