Explicit estimates for the sum ∑nk=0k!(nk)2(−1)k

We are interested in finding an explicit estimate to the binomial sum Qn(x)=∑nk=0k!(nk)2(−x)k at x=1 for n=0,1,2,…. Despite of its own interest the polynomial Qn(x) is important as the denominator in the Padé identity of the Euler's factorial series E(x)=∑∞k=0k!xk as well as its close connection to a classical Laguerre polynomial Ln(x)=1n!ex(ddx)n(e−xxn). Our main result is the explicit bound

∣∣∣Ln(1)−eπ−−√⋅cos(2n−−√−π4)n1/4+1748eπ−−√sin(2n−−√−π4)n3/4∣∣∣<0.51n

for all n=0,1,2,…, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound |Qn(1)|≤n!, and even more.