A334641 a(n) is the total number of down steps between the 3rd and 4th up steps in all 2-Dyck paths of length 3*n.

0, 0, 0, 43, 108, 444, 2099, 10683, 56994, 314296, 1776519, 10236081, 59892690, 354886920, 2125117332, 12839859620, 78176677734, 479177993904, 2954360065247, 18309779343549, 114001476318240, 712751759478780, 4472908385838795, 28165267333869435

Offset: 0

Author

Benjamin Hackl, May 07 2020

Keywords

• nonn
• easy

Comments

A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

For n = 3, there is no 4th up step, a(3) = 43 enumerates the total number of down steps between the 3rd up step and the end of the path.

Links

• Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Crossrefs

Cf. A007226, A007228, A124724, A334640, A334642, A334682.

Programs

Mathematica

a[0] = a[1] = a[2] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)

PARI

a(n) = if (n<=2, 0, 2*sum(j=1, 3, binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

Formula

a(0) = a(1) = a(2) = 0 and a(n) = 2*Sum_{j=1..3}binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)) for n > 2.