A334608 a(n) is the total number of down-steps after the final up-step in all 3_1-Dyck paths of length 4*n (n up-steps and 3n down-steps).

0, 5, 34, 236, 1714, 12922, 100300, 796572, 6443536, 52909593, 439896626, 3695917940, 31331587252, 267669458420, 2302188456120, 19918434257052, 173240112503520, 1513821095788420, 13283883136738344, 117009704490121520, 1034217260142108570, 9169842145476773250, 81537271617856588380

Offset

0,2

Comments

A 3_1-Dyck path is a lattice path with steps U=(1, 3), d=(1, -1) that starts at (0,0), stays (weakly) above y=-1, and ends at the x-axis.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1027

Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020-2022.

Paul Drube, Raised k-Dyck paths, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.

Formula

a(n) = 2*binomial(4*(n+1)+2, n+1)/(4*(n+1)+2) - 4*binomial(4*n+2, n)/(4*n+2).

a(n) ~ 101 * 2^(8*n+9/2) / (3^(3*n+11/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 26 2025

Example

For n=1, a(1)=5 is the total number of down-steps after the last up-step in Uddd, dUdd.

Mathematica

a[n_] := 2 * Binomial[4*n + 6, n + 1]/(4*n + 6) - 4 * Binomial[4*n + 2, n]/(4*n + 2); Array[a, 23, 0] (* Amiram Eldar, May 13 2020 *)

Prog

(SageMath) [2*binomial(4*(n + 1) + 2, n + 1)/(4*(n + 1) + 2) - 4*binomial(4*n + 2, n)/(4*n + 2) for n in srange(30)] # Benjamin Hackl, May 13 2020

Crossrefs

Cf. A002293, A007226, A007228, A334609, A334645, A334646, A334647, A334648, A334649, A334680, A334682, A334785.

Sequence in context: A102436 A291027 A033889 * A120469 A180909 A183415

Adjacent sequences: A334605 A334606 A334607 * A334609 A334610 A334611

Keyword

nonn,easy

Author

Andrei Asinowski, May 13 2020

Status

approved