OEIS A000321: Hermite polynomials evaluated at 12

We examine A000321, the sequence obtained by evaluating the physicist's Hermite polynomials H_n(-1/2), where H_n(x) . It comes from a compact recurrence and exhibits a modular pattern: A_{n+k} ≡ c(n,k) A_n (mod k), revealing hidden structure beneath the apparent chaos. Hermite polynomials are central in physics (quantum harmonic oscillator), appear in probability (Edgeworth expansions), Brownian motion, and random matrix theory, and even feature in signal processing as Hermite wavelets. This single sequence shows how a simple rule from one area threads through number theory, physics, and engineering.

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