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A335340 North-East paths from (0,0) to (n,n) with k cyclic descents. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..55.

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

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

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A103371.

Sequence in context: A285702 A104000 A199943 * A013599 A276528 A205839

Adjacent sequences:  A335337 A335338 A335339 * A335341 A335342 A335343

KEYWORD

nonn,tabl

AUTHOR

Per W. Alexandersson, Jun 02 2020

STATUS

approved

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Last modified October 19 21:54 EDT 2021. Contains 348095 sequences. (Running on oeis4.)