OEIS A000178: The Superfactorial

In this episode we explore OEIS sequence A000178, the superfactorial. We define it in two equivalent ways: as the product 1! · 2! · 3! · ... · n! and as the product 1^n · 2^{n-1} · 3^{n-2} · ... · n^1, noting that the case n = 0 yields 1. We look at the first terms 1, 1, 2, 12, 288, 34560 and touch on older indexing names like M2049 and N0A11. The discussion then dives into deep connections across math: its appearance as a Vandermonde determinant (the determinant of the matrix with entries i^j for i = 1..n+1 and j = 0..n equals the superfactorial), and related determinant formulations involving Lucas sequences; odd-indexed superfactorials arising as Hankel transforms of tangent numbers; determinant representations tied to Lucas-based matrices; a link to graph theory via the multiplicative Wiener index of a path graph; the striking fact that, except for n = 0 and 1, superfactorials are never perfect squares; and the Barnes G function with A_n = G(n+2), plus asymptotics related to the Glaisher–Kinkelin constant. We’ll also reflect on how these diverse threads illustrate the rich interconnectedness of algebra, combinatorics, and number theory in the OEIS landscape.

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