We unpack A00312, the self-exponentiating sequence n^n. Discover what it counts—endofunctions on an n-element set—along with its appearances as n-by-n 0-1 matrices with one 1 per row and as the count of length-n words over an n-letter alphabet. We’ll also explore the elegant base-n representation “1 followed by n zeros,” connect to probabilistic ideas like fair dice games, and glimpse the broader links to statistics and modern algebra, illustrating how a simple definition reveals a vast mathematical landscape.

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