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

%I A335340 #13 Feb 09 2021 21:00:34
%S A335340 2,4,2,6,12,2,8,36,24,2,10,80,120,40,2,12,150,400,300,60,2,14,252,
%T A335340 1050,1400,630,84,2,16,392,2352,4900,3920,1176,112,2,18,576,4704,
%U A335340 14112,17640,9408,2016,144,2,20,810,8640,35280,63504,52920,20160,3240,180,2
%N A335340 North-East paths from (0,0) to (n,n) with k cyclic descents.
%C A335340 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.
%C A335340 The sum of the entries in row n is equal to binomial(2n,n).
%C A335340 I conjecture that the polynomial Sum_{k=1...n} T(n,k) t^k is real-rooted for all n.
%H A335340 Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, <a href="https://arxiv.org/abs/2010.11157">Refined Catalan and Narayana cyclic sieving</a>, arXiv:2010.11157 [math.CO], 2020.
%F A335340 T(n,k) = 2*binomial(n,k)*binomial(n-1,k-1).
%F A335340 T(n,k) = 2 * A103371(n-1,k-1). - _Alois P. Heinz_, Jun 02 2020
%e A335340 The table starts as
%e A335340 2,
%e A335340 4, 2
%e A335340 6, 12, 2
%e A335340 8, 36, 24, 2
%e A335340 10, 80, 120, 40, 2
%e A335340 12, 150, 400, 300, 60, 2
%t A335340 T[n_, k_] = 2 Binomial[n, k] Binomial[n - 1, k - 1];
%Y A335340 Cf. A103371.
%K A335340 nonn,tabl
%O A335340 1,1
%A A335340 _Per W. Alexandersson_, Jun 02 2020