Dive into A000111, the Euler up-down numbers counting alternating permutations (up-down and down-up). We'll uncover how Andre linked them to trigonometry via a recurrence, explore their surprising ties to Bernoulli numbers, increasing 012 trees, and tangent/second numbers, and walk through the explicit Stirling-number–based formula and the pi-involving asymptotics that govern their growth. A journey through combinatorics, trig, and number theory that reveals why one simple zigzag sequence threads through so many areas.

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