A368379 Arises from enumeration of a certain class of partial zig-zag knight's paths on the square grid.

0, 1, 0, 3, 1, 9, 6, 28, 27, 90, 110, 297, 429, 1001, 1638, 3432, 6188, 11934, 23256, 41990, 87210, 149226, 326876, 534888, 1225785, 1931540, 4601610, 7020405, 17298645, 25662825, 65132550, 94287120, 245642760, 347993910, 927983760, 1289624490, 3511574910

Offset: 0

Author

N. J. A. Sloane, Feb 18 2024

Keywords

• nonn

Comments

It would be nice to have a more precise definition.

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 3.2.

Crossrefs

Cf. A368378, A368380.

The two bisections are A000245 and A115145 (shifted, negated).

Programs

Mathematica

r = (1 - 2z^2 - Sqrt[1-4z^2]) / (2z^2);
gf = (r^2 z + r u^2 + r u + 2 r z + z) / (z (1 - r u));
Table[SeriesCoefficient[gf,{u,0,2},{z,0,n}], {n,0,50}] (* Andrei Zabolotskii, Jul 25 2025 *)

Formula

G.f.: (1/x + 1 + 2*R(x) + R(x)^2) * R(x)^2 + R(x) / x = (F(x) + 1/x) * R(x), where R(x) = (1 - sqrt(1-4*x^2)) / (2*x^2) - 1 and F(x) is the g.f. of A368378. - Andrei Zabolotskii, Jul 25 2025

Extensions

Terms a(11) and beyond from Andrei Zabolotskii, Jul 25 2025