login
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
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
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: A371064 A104000 A199943 * A013599 A276528 A205839
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved