Eulerian Numbers

A(n,k) = permutations of n elements with exactly k descents

Highlight row n=

A(n,k) = (k+1)·A(n−1,k) + (n−k)·A(n−1,k−1) | Worpitzky: x^n = Σ A(n,k)·C(x+k,n)

Each row sums to n! (all permutations). The Worpitzky identity expresses powers of x as a sum weighted by Eulerian numbers. Row 4: [1,11,11,1] — the "Eulerian palindrome."