A334643 a(n) is the total number of down steps between the second and third up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

0, 0, 16, 53, 209, 963, 4816, 25367, 138531, 777041, 4449511, 25901655, 152818458, 911755012, 5491420104, 33343242196, 203881825163, 1254342228285, 7759025239189, 48227078649155, 301056318504165, 1886647802277315, 11864793375611820, 74854437302309175

Offset: 0

Author

Benjamin Hackl, May 12 2020

Keywords

• nonn
• easy

Comments

For n = 2, there is no 3rd up step, a(2) = 16 enumerates the total number of down steps between the 2nd up step and the end of the path.

Examples

			For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 4 + 3 + 2 + 1 + 1 + 2 + 3 = 16 down steps between the 2nd up step and the end of the path.
		

Links

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

Crossrefs

Cf. A001764, A007226, A030983, A334640, A334642, A334644.

Programs

SageMath

[binomial(3*n + 1, n)/(3*n + 1) + 4*sum([binomial(3*j + 2, j)*binomial(3*(n - j), n - j)/(3*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 7*(n==2) if n >= 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

Formula

a(0) = a(1) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..2}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 7*[n=2] for n > 1, where [ ] is the Iverson bracket.