A334650 a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.

0, 6, 31, 158, 975, 6639, 48050, 362592, 2820789, 22460120, 182141553, 1499143282, 12490923757, 105150960654, 892973346300, 7640934031920, 65813450140017, 570160918044288, 4964875184429660, 43431741548248440, 381496856026500220, 3363457643008999635

Offset

0,2

Comments

A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

For n = 1, there is no 2nd up step, a(1) = 6 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..21.

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) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) + 9*binomial(4*(n-1), n-1)/n - 6*[n=1] for n > 0, where [ ] is the Iverson bracket.

Example

For n = 1, the 3_2-Dyck paths are DDUD, DUDD, UDDD. This corresponds to a(1) = 1 + 2 + 3 = 6 down steps between the 1st up step and the end of the path.

Mathematica

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - Binomial[4*n + 2, n]/(n + 1) + 9 * Binomial[4*(n - 1), n - 1]/n - 6 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 13 2020 *)

Prog

(SageMath) [3*binomial(4*n, n)/(n + 1) - binomial(4*n + 2, n)/(n + 1) + 9*binomial(4*(n - 1), n - 1)/n - 6*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 13 2020

Crossrefs

Cf. A001764, A007226, A007228, A334609, A334647, A334648, A334649, A334785.

Sequence in context: A065096 A077352 A038223 * A022034 A277669 A047665

Adjacent sequences: A334647 A334648 A334649 * A334651 A334652 A334653

Keyword

nonn,easy

Author

Benjamin Hackl, May 13 2020

Status

approved