OEIS A000318: Generalized tangent numbers d(4,n)

In this Deep Dive, we explore OEIS A000318, the generalized tangent numbers, often denoted d(4,N). The initial terms—4, 128, 16384—hint at incredibly rapid growth, and the sequence sits at a rich crossroads of history, combinatorics, and analysis. We'll trace its origins in Sloan’s 1973 handbook and its later entry in the OEIS (1995), and unpack the explicit link to A000182—Euler-type numbers—in a precise formula. We’ll see how these numbers are the coefficients in the Maclaurin expansion of tan(4x), connect that generating function to a continued fraction, and understand how the convergents are Padé approximants to the tan(4x) series, bridging discrete sequences with analytic structure. Finally, we’ll reflect on the OEIS keywords—nonnegative and easy—and what they reveal about the hidden depth behind a simple integer sequence.

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