Explicit estimates for the sum ${\sum }_{k=0}^{n}k!{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2(-1{)}^k$

Anne-Maria Ernvall-Hytönen, Tapani Matala-aho

We are interested in finding an explicit estimate to the binomial sum $Q_n(x)={\sum }_{k=0}^{n}k!{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2(-x{)}^k$ at $x=1$ for $n=0,1,2,\dots$. Despite of its own interest…