Let $k\ge n$ and $$A(k,n)=\frac{ \Gamma[1+k]}{n!\Gamma[1+k-n]^2}\int_0^\infty \frac{e^{-r}r^{k-n}}{L_n(-r)} dr$$
where $$L_n(-r) = \sum_{m=0}^n \frac{\Gamma(1+n)}{\Gamma(1+m)^2 \Gamma(1+n-m)}r^m$$ is the is Laguerre polynomial.
Then it seems that $A(k,n)<1$, but still no proof. This can be proved for several integers $n=2,3..$, for $k\ge n$, but it is not easy for general n. It is enough to prove the monotonity of $A$ w.r.t $n$ or $k$.
