A334976 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative path with steps (1, 2), (1, -1) that starts and ends at y = 0.

0, 0, 3, 19, 108, 609, 3468, 20007, 116886, 690690, 4122495, 24823188, 150629248, 920274804, 5656456104, 34954487967, 217044280458, 1353539406660, 8474029162305, 53241343026795, 335592121524660, 2121577490385885, 13448859209014320, 85467026778421860

Offset

0,3

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1211

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(0) = 0 and a(n) = binomial(3*n+4, n+1)/(3*n+4) - 3*binomial(3*n+1, n)/(3*n+1) for n > 0.

a(n) ~ 5 * 3^(3*n+3/2) / (2^(2*n+4) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 23 2025

Example

For n = 2, the 2-Dyck paths are UDDUDD, UDUDDD, UUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 2+1+0 = 3.

Maple

alias(PS=ListTools:-PartialSums): A334976List := proc(m) local A, P, n;
A := [0, 0]; P := [1, 0]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334976List(24); # Peter Luschny, Mar 26 2022

Mathematica

a[0] = 0; a[n_] := Binomial[3*n+4, n+1]/(3*n + 4) - 3 * Binomial[3*n + 1, n]/(3*n + 1); Array[a, 24, 0]

Prog

(SageMath) [binomial(3*n + 4, n + 1)/(3*n + 4) - 3*binomial(3*n + 1, n)/(3*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Crossrefs

Cf. A334977, A334978, A334979, A334980.

Sequence in context: A072950 A240123 A130425 * A103005 A162354 A132959

Adjacent sequences: A334973 A334974 A334975 * A334977 A334978 A334979

Keyword

nonn,easy

Author

Sarah Selkirk, May 18 2020

Status

approved