A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3*n (n up-steps and 2*n down-steps).

0, 2, 9, 43, 218, 1155, 6324, 35511, 203412, 1184040, 6983925, 41652468, 250763464, 1521935948, 9301989144, 57203999295, 353701790376, 2197600497330, 13713291247635, 85907187607395, 540072341320050, 3406202392821375, 21545888897092560, 136655834260685220, 868897745157965328

Offset: 0

Author

Andrei Asinowski, May 08 2020

Keywords

• nonn
• easy

Comments

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

Examples

			For n = 2, the a(2) = 9 is the total number of down-steps after the last up-step in UddUdd, UdUddd, UUdddd.
		

Links

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

Crossrefs

First order differences of A001764.

The 4th column of A280759.

Cf. A062745.

Programs

Maple

alias(PS=ListTools:-PartialSums): A334680List := proc(m) local A, P, n;
A := [0,2]; P := [1,2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334680List(25); # Peter Luschny, Mar 26 2022

Mathematica

a[n_] := Binomial[3*n + 4, n + 1]/(3*n + 4) - Binomial[3*n + 1, n]/(3*n + 1); Array[a, 25, 0] (* Amiram Eldar, May 13 2020 *)

SageMath

[(17 + 23*n)*binomial(3*n, n-1)/(2*n+2)/(2*n+3) for n in srange(30)] # Benjamin Hackl, May 13 2020

Formula

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

a(n) = (17 + 23*n)*binomial(3*n, n - 1)/((2*n + 2)*(2*n + 3)).

a(n) = A062745(n+1, 2*n-1).