OEIS A000153: A bead-and-necklace counting recurrence

In this episode we unpack A000153, the recurrence a(n) = n a(n−1) + (n−2) a(n−2) with a(0)=0 and a(1)=1. We’ll explore its rapid growth and a bead-on-necklace interpretation with two indistinguishable cords, examine connections to subfactorials (A000166), inverse binomial transforms, and matrix permanents, and peek at hypergeometric, gamma, and generating-function formulas that link this sequence to broad areas of combinatorics and analysis.

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