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A368375 Arises from enumeration of a certain class of partial knight's paths on the square grid.

%I A368375 #14 Jul 26 2025 10:21:22
%S A368375 0,1,0,2,1,9,10,42,64,213,400,1180,2510,6918,15842,42134,100966,
%T A368375 263894,650688,1687746,4238855,10968196,27888906,72187356,185141385,
%U A368375 480025465,1238940890,3219591600,8350054339,21752631006,56634075418,147899071110,386293111740
%N A368375 Arises from enumeration of a certain class of partial knight's paths on the square grid.
%C A368375 It would be nice to have a more precise definition.
%H A368375 Jean-Luc Baril and José L. Ramírez, <a href="https://jl.baril.u-bourgogne.fr/knight.pdf">Knight's paths towards Catalan numbers</a>, Univ. Bourgogne Franche-Comté (2022). Also arXiv:<a href="https://arxiv.org/abs/2206.12087">2206.12087</a> [math.CO], Jan 2023. See Section 2.1.
%F A368375 G.f.: (1 - (1+x) * A(x) + (x+x^4) * A(x)^2) / (x * (x * A(x) - 1)), where A(x) is the g.f. of A005220. - _Andrei Zabolotskii_, Jul 25 2025
%t A368375 aa = (1 + 2z + Sqrt[1 - 4z + 4z^2 - 4z^4] - Sqrt[2]*Sqrt[1 - 4z^2 - 2z^4 + (2z + 1)*Sqrt[1 - 4z + 4z^2 - 4z^4]])/(4z^2);
%t A368375 a1 = z^2 aa^2 / (1 - z aa);
%t A368375 gf = (z (u z + 1) aa + u z a1 - u^2) / (u^4 z + u^3 z^2 + u z^2 - u^2 + z);
%t A368375 Table[SeriesCoefficient[gf,{u,0,2},{z,0,n}], {n,0,30}] (* _Andrei Zabolotskii_, Jul 25 2025 *)
%Y A368375 Cf. A005220, A005221.
%K A368375 nonn
%O A368375 0,4
%A A368375 _N. J. A. Sloane_, Feb 17 2024
%E A368375 Terms a(10) and beyond from _Andrei Zabolotskii_, Jul 25 2025