A368377 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.
0, 0, 0, 0, 1, 0, 2, 0, 6, 1, 15, 4, 37, 14, 91, 44, 222, 129, 541, 364, 1319, 1000, 3219, 2696, 7869, 7172, 19273, 18892, 47299, 49398, 116317, 128444, 286624, 332552, 707679, 858168, 1750588, 2208898, 4338314, 5674380, 10769893, 14554398, 26780522, 37286820
Offset: 0
Keywords
Links
- Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087 [math.CO], Jan 2023. See Section 2.2.
Programs
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Mathematica
r = (1 - z^4 - z^2 - Sqrt[z^8 - 2z^6 - z^4 - 2z^2 + 1]) / (2z^3); gf = r (u^2 z + u z^2 + 1) / (z^3 (1 - r u)); Table[SeriesCoefficient[gf,{u,0,3},{z,0,n}], {n,0,50}] (* Andrei Zabolotskii, Jul 25 2025 *)
Formula
G.f.: (x + x^2 * R(x) + R(x)^2) * R(x)^2 / x^3 = F(x) * R(x), where R(x) = x * (A(x^2) - 1), A(x) is the g.f. of A004148, and F(x) is the g.f. of A368376. - Andrei Zabolotskii, Jul 25 2025
Extensions
Terms a(16) and beyond from Andrei Zabolotskii, Jul 25 2025
Comments