Asymptotics of the Eulerian numbers revisited: A large deviation analysis

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Eulerian numbers \( A_{n,k} \) and Cauchy’s integral formula, asymptotic results in non-central region. In the region \( k = n – n^\alpha \), \( 1 > \alpha > 1/2 \), we analyze the dependence of \( A_{n,k} \) on \(\alpha\). This paper fits within the framework of Analytic Combinatorics.