  We are interested in finding an explicit estimate to the binomial sum
$Q_n(t)=\sum_{k=0}^{n} k! {n\choose k}^2 (-t)^{k}$ at $t=1$ for
$n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(t)$ is
important as the denominator in the Pad\'e identity of the Euler's factorial
series $E(t) = \sum_{k=0}^{\infty} k! t^k$ as well as its close connection to a
classical Laguerre polynomial $L_n(t) = \frac{1}{n!} e^t
\left(\frac{d}{dt}\right)^n (e^{-t}t^n)$. Our main result is the explicit bound
$\left|L_n(1)-\sqrt{\frac{e}{\pi}}\cdot
\frac{\cos\left(2\sqrt{n}-\frac{\pi}{4}\right)}{n^{1/4}}\right|<\frac{2.38}{n^{3/4}}$
for all $n=0,1,2,\ldots$, which replaces the F\'ejer's asymptotic formula from
1909. As a corollary of this, one also gets a new proof for the bound
$|Q_{n}(1)| \le n!$, and even more.

Number Theory