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Ryan J. Amaral, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/2609.27096">Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity</a>, arXiv:2609.27096 [math.CO], 2026. See pp. 1-2.

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Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, <a href="httphttps://jl.baril.u-bourgogne.fr/narayana.pdf">Generalized Narayana arrays, restricted Dyck paths, and related bijections</a>, Univ. Bourgogne (France, 2025). See p. 20.

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Chao-Jen Wang, <a href="httphttps://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf">Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords</a>.

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a(n) = -(1/2) * Sum_{k=0..n+1} (binomial(1/2, k) * binomial(-1/2, n+1-k) * (-3)^k) for n >= 0. - Dmytro Dmytryshyn, Sep 26 2026