A334978 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

0, 0, 6, 52, 409, 3208, 25484, 205452, 1679332, 13894848, 116193246, 980658172, 8343605534, 71492410640, 616418176920, 5344364518140, 46565472754044, 407529832131712, 3580911446989368, 31579384975219920, 279414033129153065, 2479725948121016040

Offset: 0

Author

Sarah Selkirk, May 18 2020

Keywords

• nonn
• easy

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 0 is the total number of down steps before the first up step.

Examples

			For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. Therefore, the total number of down steps between the first and second up steps is a(2) = 3 + 2 + 1 + 0 = 6.
		

Links

• A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Crossrefs

Cf. A334976, A334977, A334979, A334980.

Programs

Mathematica

a[0] = 0; a[n_] := Binomial[4*n+5, n+1]/(4*n + 5) - 4 * Binomial[4*n + 1, n]/(4*n + 1); Array[a, 22, 0]

SageMath

[binomial(4*n + 5, n + 1)/(4*n + 5) - 4*binomial(4*n + 1, n)/(4*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = binomial(4*n+5, n+1)/(4*n+5) - 4*binomial(4*n+1, n)/(4*n+1) for n > 0.