OEIS A000320: Generalized tangent numbers d(5,n)

Today we explore A000320, the generalized tangent numbers with d(5,n). We trace their history—from Sloan’s foundational entries and Shanks’s early notes (old IDs M3722, N5521) to Sloan’s Handbook of Integer Sequences and the 1995 encyclopedia collaboration with Simon Plouffe. The modern definition comes from Peter Lishney (2021): the a_n arise from the coefficient extraction in the power series of sec(5x)·(sin x + sin 3x), with a_n equal to (2n−1)! times the coefficient of x^{2n−1}. The sequence starts 4, 272, 55,744, 23,750,912 and grows rapidly, and OEIS cross-references link it to related sequences such as A000318 and A000187, highlighting the rich web of connections in number theory and combinatorics.

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