OEIS A000317: Quadratic recurrence and integer polynomial binomial coefficients

We explore the nonlinear recurrence A_{n+1} = A_n^2 - A_n A_{n-1} + A_{n-1}^2, tracing its explosive growth, and explain Emmanuel Ferrand’s 2007 discovery that A000317 belongs to a special class whose generalized binomial coefficients are polynomials with integer coefficients. This reveals an elegant algebraic structure beneath a rapidly growing sequence, linking the recurrence to polynomial algebra and the idea of deformations of the Taylor formula.

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