A393573 - OEIS

%I #55 Apr 10 2026 10:31:54

%S 1,1,4,1,20,21,1,88,252,121,1,376,2354,2547,728,1,1596,20249,38335,

%T 22847,4488,1,6764,167998,509199,482715,190892,28101,1,28656,1369444,

%U 6364996,8677074,5229786,1523818,177859,1,121392,11063417,76894701,143532700,118307789,51469561,11791064,1134705

%N Triangle read by rows, T(n, k) is the number of 3-dimensional balanced ballot paths of 3n steps and semisymmetric height of exactly 2k, where the semisymmetric height of point (x_1, x_2, x_3, x_4) is defined as h_3'(x) = 2*x_1 - 2*x_3 for 1 <= k <= n.

%C For point x (x_1,x_2,x_3) in the 3-dimensional lattice, we define the semisymmetric height of x as h_3'(x) = 2*x_1 - 2*x_3. The 3-dimensional balanced ballot path (multidimensional Dyck path) is a sequence of 3*n steps with initial point (0,0,0,0) and ending at (n,n,n,n) satisfying that each step is a standard unit vector and each point of the path satisfies x_1 >= x_2 >= x_3. T(n,k) is the number of 3-dimensional balanced ballot paths of 3*n steps such that the largest semisymmetric height reached by any point in the path is equal to 2k; i.e., for at least one intermediate point h_3'(x) = 2k, but for no intermediate points h_3'(x) > 2k. Here, 1 <= k <= n.

%C We call h_3' the semisymmetric height because (a) the coefficient of x_{4-i} in the formulation of h_3' in the negation of the coefficient of x_i and (b) h_3' is invariant under the function g(x_1, x_2, x_3) = (n-x_3, n-x_2, n-x_1).

%C It is important to note that, here, the semisymmetric height of a point in the 3-dimensional lattice is different from the height defined in the comments of sequence A387912.

%H Ryota Inagaki and Dimana Pramatarova, <a href="https://arxiv.org/abs/2604.04900">On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers</a>, arXiv:2604.04900 [math.CO], 2026. See p. 26.

%e Triangle begins:

%e   1;

%e   1,    4;

%e   1,   20,     21;

%e   1,   88,    252,    121;

%e   1,  376,   2354,   2547,    728;

%e   1, 1596,  20249,  38335,  22847,   4488;

%e   1, 6764, 167998, 509199, 482715, 190892, 28101;

%o (Python)

%o from functools import lru_cache

%o @lru_cache(None)

%o def ht(t):

%o     return 2 * t[0] - 2 * t[2]

%o def f(t, bd):

%o     count = 0

%o     if t == (0, 0, 0):

%o         return 1

%o     if t[0] > 0 and t[0] - 1 >= t[1]:

%o         count += f((t[0] - 1, t[1], t[2]), bd)

%o     if t[1] > 0 and t[1] - 1 >= t[2] and ht((t[0], t[1] - 1, t[2])) <= bd:

%o         count += f((t[0], t[1] - 1, t[2]), bd)

%o     if t[2] > 0 and ht((t[0], t[1], t[2] - 1)) <= bd:

%o         count += f((t[0], t[1], t[2] - 1), bd)

%o     return count

%o for n in range(1, 9):

%o     for h in range(1, n+1):

%o         if h == 1:

%o             print(f((n, n, n), 2*h))

%o         if h > 1:

%o             print(f((n, n, n), 2*h)-f((n, n, n), 2*(h-1)))

%Y Cf. A393571, A393594, A387912.

%Y Row sums give A005789.

%K nonn,tabl

%O 1,3

%A _Ryota Inagaki_ and _Dimana Pramatarova_, Feb 21 2026