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A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.

%I A332389 #21 Feb 11 2020 12:05:27
%S A332389 1,1,2,1,4,3,1,8,7,4,1,16,18,10,5,1,32,47,28,13,6,1,64,123,82,38,16,7,
%T A332389 1,128,322,244,117,48,19,8,1,256,843,730,370,152,58,22,9,1,512,2207,
%U A332389 2188,1186,496,187,68,25,10,1,1024,5778,6562,3827,1648,622,222,78,28,11
%N A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.
%C A332389 A(n,w) is the number of circular Dyck paths of size n, and width at most w.
%C A332389 This is also the number of circular area lists, a_1, a_2, ..., a_n such that 0 <= a_i <= w-1, and a_{i+1} < a_i + 1, for all 1 <= i <= n, and the index i is taken modulo n.
%C A332389 The values of w are given by the row index.
%C A332389 A(n,w) is given by summing binomial(2*n - 1, n - 1 - (w+2) k) - binomial(2*n - 1, n + j + (w+2)*k) over k=1..w and k over all integers.
%H A332389 Per Alexandersson, Svante Linusson and Samu Potka, <a href="https://arxiv.org/abs/1903.01327">The cyclic sieving phenomenon on circular Dyck paths</a>, arXiv:1903.01327 [math.CO], 2019.
%H A332389 Per Alexandersson, Svante Linusson and Samu Potka, <a href="https://doi.org/10.37236/8720">The cyclic sieving phenomenon on circular Dyck paths</a>, Electronic Journal of Combinatorics 26, No.4 (2019).
%F A332389 A(n,w) = Sum_{k=-2*(n+2)..2*(n+2)} Sum_{j=1..w} binomial(2n-1, n-1-(w+2)*k) - binomial(2*n-1, n + j + (w+2)*k).
%e A332389 The table begins as
%e A332389 1,    2,    3,    4,    5, ...
%e A332389 1,    4,    7,    10,   13, ...
%e A332389 1,    8,    18,   28,   38, ...
%e A332389 1,    16,   47,   82,   117, ...
%e A332389 1,    32,   123,  244,  370, ...
%e A332389 ...
%e A332389 A(5,3)=123 and a few of the corresponding circular area lists are 00000, 10000,...,12210,...,12222, 22222.
%t A332389 CircularDyckPaths[n_, w_] := With[{d = w + 2},
%t A332389    Sum[Binomial[2 n - 1, n - 1 - d s] -
%t A332389      Binomial[2 n - 1, n + j + d s]
%t A332389     , {j, w},
%t A332389     {s, -2 (n + 2), 2 (n + 2)}]
%t A332389    ];
%t A332389 Table[
%t A332389 CircularDyckPaths[n, w]
%t A332389 , {n, 1, 10}, {w, 1, 10}]
%Y A332389 A194460 is the diagonal.
%K A332389 nonn,tabl
%O A332389 1,3
%A A332389 _Per W. Alexandersson_, Feb 10 2020