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Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L. (2018). Enumerations of peaks and valleys on non-decreasing Dyck paths. Discrete Mathematics, 341(10), 2789-2807. See Table 4.
E. Czabarka et al, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Disc. Math. 341 (2018) 2789-2807. See Table 4.
G.f.: x(1-x)/((1-2x)(1-3x+x^2)).
G.f.: x(1-x)/((1-2x)(1-3x+x^2)); a(n) = sum{k=0..n+1, binomial(n+1, k+1)*sum{j=0..floor(k/2), F(k-2j)}}.
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Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 22.
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(MAGMAMagma) [Fibonacci(2*n+2)-2^n: n in [0..30]]; // Vincenzo Librandi, Apr 21 2011
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