A393573 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.

1, 1, 4, 1, 20, 21, 1, 88, 252, 121, 1, 376, 2354, 2547, 728, 1, 1596, 20249, 38335, 22847, 4488, 1, 6764, 167998, 509199, 482715, 190892, 28101, 1, 28656, 1369444, 6364996, 8677074, 5229786, 1523818, 177859, 1, 121392, 11063417, 76894701, 143532700, 118307789, 51469561, 11791064, 1134705

Offset

1,3

Comments

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.

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).

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.

Links

Table of n, a(n) for n=1..45.

Ryota Inagaki and Dimana Pramatarova, On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers, arXiv:2604.04900 [math.CO], 2026. See p. 26.

Example

Triangle begins:
  1;
  1,    4;
  1,   20,     21;
  1,   88,    252,    121;
  1,  376,   2354,   2547,    728;
  1, 1596,  20249,  38335,  22847,   4488;
  1, 6764, 167998, 509199, 482715, 190892, 28101;

Prog

(Python)
from functools import lru_cache
@lru_cache(None)
def ht(t):
    return 2 * t[0] - 2 * t[2]
def f(t, bd):
    count = 0
    if t == (0, 0, 0):
        return 1
    if t[0] > 0 and t[0] - 1 >= t[1]:
        count += f((t[0] - 1, t[1], t[2]), bd)
    if t[1] > 0 and t[1] - 1 >= t[2] and ht((t[0], t[1] - 1, t[2])) <= bd:
        count += f((t[0], t[1] - 1, t[2]), bd)
    if t[2] > 0 and ht((t[0], t[1], t[2] - 1)) <= bd:
        count += f((t[0], t[1], t[2] - 1), bd)
    return count
for n in range(1, 9):
    for h in range(1, n+1):
        if h == 1:
            print(f((n, n, n), 2*h))
        if h > 1:
            print(f((n, n, n), 2*h)-f((n, n, n), 2*(h-1)))

Crossrefs

Cf. A393571, A393594, A387912.

Row sums give A005789.

Sequence in context: A144354 A049352 A322218 * A385582 A182867 A382206

Adjacent sequences: A393570 A393571 A393572 * A393574 A393575 A393576

Keyword

nonn,tabl

Author

Ryota Inagaki and Dimana Pramatarova, Feb 21 2026

Status

approved