Several problems in Cody ask us to construct part or all of triangles in which entries follow a pattern. Cody Problems 37, 1463, 44037, and 44904 involve Pascal's triangle, which consists of the binomial coefficients, and Cody Problem 1845 extends Pascal's triangle to a pyramid. Cody Problem 45460 involves the Bernoulli triangle, which consists of partial sums of the binomial coefficients.
This problem deals with the Seidel-Entringer-Arnold triangle (also called the Euler-Bernoulli triangle and the secant-tangent triangle). The first eight layers are
1
0 1
1 1 0
0 1 2 2
5 5 4 2 0
0 5 10 14 16 16
61 61 56 46 32 16 0
0 61 122 178 224 256 272 272
The name "secant-tangent triangle" arises because the sides contain the coefficients in the Taylor series for sec(x) and tan(x):
sec(x) = 1 + 1x^2/2! + 5x^4/4! + 61x^6/6! + ...
tan(x) = 1x + 2x^3/3! + 16x^5/5! + 272x^7/7! + ...
Construct the nth layer of this triangle.
Hint: Use the boustrophedon (or ox-plowing) rule.
Solve

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Last Solution submitted on Apr 20, 2026

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George Berken Mehdi Dehghan Hong Son Lorenzo

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