A334609 a(n) is the total number of down-steps after the final up-step in all 3_2-Dyck paths of length 4*n (n up-steps and 3*n down-steps).

0, 6, 46, 339, 2553, 19723, 155805, 1253931, 10249096, 84864051, 710429304, 6003238901, 51140131770, 438729741450, 3787208722815, 32871470376123, 286706337100656, 2511620756461504, 22089299382478728, 194966351598215340, 1726424465382128205

Offset: 0

Author

Andrei Asinowski, May 13 2020

Keywords

• nonn
• easy

Comments

A 3_2-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.

Examples

			For n = 1, a(1) = 6 is the total number of down-steps after the last up-step in Uddd, dUdd, ddUd.
		

Links

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

Crossrefs

Cf. A334785, A334650, A334682, A334608.

Programs

Mathematica

a[n_] := 3 * Binomial[4*n + 7, n + 1]/(4*n + 7) - 9 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

SageMath

[3*binomial(4*(n + 1) + 3, n + 1)/(4*(n + 1) + 3) - 9*binomial(4*n + 3, n)/(4*n + 3) for n in srange(30)] # Benjamin Hackl, May 13 2020

Formula

a(n) = 3*binomial(4*(n+1) + 3, n+1)/(4*(n+1) + 3) - 9*binomial(4*n+3, n)/(4*n + 3).