A334651 a(n) is the total number of down steps between the first and second up steps in all 4_1-Dyck paths of length 5*n.

0, 7, 25, 155, 1195, 10282, 94591, 910480, 9054965, 92310075, 959473878, 10129715890, 108327387675, 1170975480360, 12773887368040, 140445927510832, 1554748206904325, 17314584431331025, 193849445090545875, 2180550929942519685, 24632294533221865028

Offset

0,2

Comments

A 4_1-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 = -1.

For n = 1, there is no 2nd up step, a(1) = 7 enumerates the total number of down steps between the 1st up step and the end of the path.

Links

Table of n, a(n) for n=0..20.

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) - 3*binomial(5*n+1, n)/(n+1) + 8*binomial(5*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.

Example

For n = 1, the 4_1-Dyck paths are DUDDD, UDDDD. This corresponds to a(1) = 3 + 4 = 7 down steps between the 1st up step and the end of the path.

Mathematica

a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n + 1) - 3 * Binomial[5*n + 1, n]/(n + 1) + 8*Binomial[5*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

Prog

(SageMath) [4*binomial(5*n, n)/(n + 1) - 3*binomial(5*n + 1, n)/(n + 1) + 8*binomial(5*(n - 1), n - 1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)]

Crossrefs

Cf. A002294, A124724, A334642, A334647, A334719, A334786, A334787.

Sequence in context: A241714 A151491 A208425 * A366941 A191237 A088009

Adjacent sequences: A334648 A334649 A334650 * A334652 A334653 A334654

Keyword

nonn,easy

Author

Benjamin Hackl, May 13 2020

Status

approved