OEIS A00187: Generalized Euler numbers

In this Deep Dive, we explore the generalized Eulerian numbers behind OEIS A00187. We sketch what they count (a broad generalization of ascent statistics in permutations), how an inclusion–exclusion approach yields explicit formulas, and how Euler’s generating-function viewpoint connects to modern compact generating functions. We highlight a key recurrence that ties A(n,k) to earlier terms, the symmetry obtained by reading permutations in reverse, and the role of limiting Eulerian numbers as signed relatives of the classical ones. We also touch on the associated polynomial family GR_n^(r)(X) and how these ideas weave together to illuminate the rich combinatorial structure underlying generalized Eulerian numbers.

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