A335340 North-East paths from (0,0) to (n,n) with k cyclic descents.

2, 4, 2, 6, 12, 2, 8, 36, 24, 2, 10, 80, 120, 40, 2, 12, 150, 400, 300, 60, 2, 14, 252, 1050, 1400, 630, 84, 2, 16, 392, 2352, 4900, 3920, 1176, 112, 2, 18, 576, 4704, 14112, 17640, 9408, 2016, 144, 2, 20, 810, 8640, 35280, 63504, 52920, 20160, 3240, 180, 2

Offset: 1

Author

Per W. Alexandersson, Jun 02 2020

Keywords

• nonn
• tabl

Comments

A North-East path is a path from (0,0) to (n,n) using steps (1,0) and (0,1). A cyclic descent is a North step followed by an East step, where the last and first step is a cyclic descent if the path ends with a North step and starts with an East step.

The sum of the entries in row n is equal to binomial(2n,n).

I conjecture that the polynomial Sum_{k=1...n} T(n,k) t^k is real-rooted for all n.

Examples

			The table starts as
2,
4, 2
6, 12, 2
8, 36, 24, 2
10, 80, 120, 40, 2
12, 150, 400, 300, 60, 2
		

Links

• Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.

Crossrefs

Cf. A103371.

Programs

Mathematica

T[n_, k_] = 2 Binomial[n, k] Binomial[n - 1, k - 1];

Formula

T(n,k) = 2*binomial(n,k)*binomial(n-1,k-1).

T(n,k) = 2 * A103371(n-1,k-1). - Alois P. Heinz, Jun 02 2020