A368378 Arises from enumeration of a certain class of partial zig-zag knight's paths on the square grid.

0, 1, 1, 2, 4, 5, 14, 14, 48, 42, 165, 132, 572, 429, 2002, 1430, 7072, 4862, 25194, 16796, 90440, 58786, 326876, 208012, 1188640, 742900, 4345965, 2674440, 15967980, 9694845, 58929450, 35357670, 218349120, 129644790, 811985790, 477638700, 3029594040

Offset

0,4

Comments

It would be nice to have a more precise definition.

Links

Table of n, a(n) for n=0..36.

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.

Formula

G.f.: (1/x + 1 + 2*R(x) + R(x)^2) * R(x), where R(x) = (1 - sqrt(1-4*x^2)) / (2*x^2) - 1. - Andrei Zabolotskii, Jul 25 2025

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, 1}, {z, 0, n}], {n, 0, 50}] (* Andrei Zabolotskii, Jul 25 2025 *)

Crossrefs

Cf. A368379, A368380.

The two bisections are A000108 and A099376. The first differences are A026008.

Sequence in context: A104549 A174513 A000063 * A367298 A039574 A182375

Adjacent sequences: A368375 A368376 A368377 * A368379 A368380 A368381

Keyword

nonn

Author

N. J. A. Sloane, Feb 18 2024

Extensions

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

Status

approved