OEIS A000364: Euler numbers, secant numbers, and zigzag permutations

We explore A000364, the even-indexed Euler numbers (secant numbers) that count alternating permutations of even size starting with a descent. Learn how the full Euler numbers split into secant and tangent parts via Andre’s generating function sec(x) + tan(x), so sec(x) yields the down-up-down-up permutations and tan(x) the up-down-starting ones (A000182). We discuss Seidel’s triangle (the Boostrifidon Transform) for efficient computation, the fact that all A000364 terms are odd, and their relation to hyperbolic secant. Finally, we connect the growth of these counts to the nearest singularity of sec x and tan x, revealing a surprising link to pi and the analytic side of a combinatorial problem.

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