I am looking for a source of the
result that the largest Eulerian numbers
$A(n,m)$ for a fixed $n$ are the central ones.
Specifically, for $n$ odd, $A(n,m)$ is largest
for $m=\frac{1}{2}(n-1)$ and, for $n$ even,
$A(n,\frac{n}{2}-1)=A(n,\frac{n}{2})$ is maximal.
It's not clear to me how to use the closed-form expression
$A(n,m) = \sum_{j=0}^{m+1}(-1)^j{n+1 \choose j}(m+1-j)^n$
or the recurrence
$A(n,m) = (n-m)A(n-1,m-1) + (m+1)A(n-1,m)$
with $A(1,0)=1$ to establish the result. I have found the paper by Lesieur and Nicholas in Europ. J. Combinatorics (1992) 13, 379-399. The paragraph after the table on page 379 uses what appears to be a well-known fact that is unproven in the paper about Eulerian numbers increasing and then decreasing for a fixed $n$.
Thank you @Phicar. R. P. Stanley's paper Log-Concave and Unimodal in Algebra, Combinatorics, and Geometry covers this topic and cites L. Comtet's book Advanced Combinatorics.