OEIS A000295: The one-descent Eulerian numbers (and why they’re not Euler’s triangle)

In this episode we untangle OEIS A000295, the sequence that appears with Eulerian-number flavor but isn’t the classic geomet rics Euler triangle. The entries 0, 0, 1, 4, 11, 26, 57 follow the closed form a(n) = 2^n − n − 1 for n ≥ 0. We’ll explore two equivalent combinatorial interpretations: - a(n) counts permutations of {1,...,n} that have exactly one descent; and - a(n) counts the nonempty subsets of an n-element set that have size at least 2.For example, n = 3 gives a(3) = 4 and n = 4 gives a(4) = 11.We’ll briefly connect this to Eulerian numbers A(n,k) with k = 1, and explain how a single sequence can arise from a simple triangle-like counting rule.Next, we pivot to the geometric side often associated with Euler: the Euler triangle and the Euler line. This is a separate concept from the OEIS sequence. The geometric Euler triangle is formed using the orthocenter, and its associated nine-point circle and the Euler line tie together orthocenter, circumcenter, centroid, and nine-point center. We’ll outline the key facts and the special cases (right and isosceles triangles) that make this a striking geometric structure.Bottom line: two very different Euler-related ideas share a name but belong to different branches of math—one a tidy combinatorial count, the other a rich geometric configuration. Both testify to Euler’s deep influence across mathematics.

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