A334787 a(n) is the total number of down steps before the first up step in all 4_3-Dyck paths of length 5*n. A 4_3-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -3.

0, 6, 34, 251, 2105, 19040, 181076, 1784728, 18067803, 186754590, 1962728460, 20910164730, 225308533359, 2451112021568, 26885549373440, 297008527319440, 3301615350645935, 36903975448964670, 414518195957729886, 4676429192392769805, 52965796433899543810

Offset

0,2

Links

Stefano Spezia, Table of n, a(n) for n = 0..900

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

Formula

a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - binomial(5*n+3, n)/(n+1) for n > 0.

a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (131/128)*sqrt(5/(2*Pi)). - Stefano Spezia, Oct 19 2022

Example

For n = 1, there are the 4_3-Dyck paths UDDDD, DUDDD, DDUDD, DDDUD. Before the first up step there are a(1) = 0 + 1 + 2 + 3 = 6 down steps in total.

Mathematica

a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - Binomial[5*n+3, n]/(n+1); Array[a, 21, 0]

Crossrefs

Cf. A001764, A002293, A002294, A334785, A334786.

Sequence in context: A230331 A267242 A197436 * A302148 A218685 A108432

Adjacent sequences: A334784 A334785 A334786 * A334788 A334789 A334790

Keyword

nonn,easy

Author

Sarah Selkirk, May 11 2020

Status

approved