A393594 Triangle read by rows: T(n, k) is the number of 4-dimensional balanced ballot paths of 4n steps and symmetric height exactly k.

1, 1, 0, 1, 8, 4, 1, 0, 3, 69, 48, 30, 151, 135, 25, 1, 0, 7, 533, 553, 583, 4299, 5051, 1930, 4288, 4819, 1764, 196, 1, 0, 15, 3990, 5835, 8727, 98724, 155043, 83355, 273825, 372281, 183904, 164508, 189360, 99044, 22428, 1764, 1, 0, 31, 29633, 58909, 117079, 2096755, 4441117, 3049235, 13012776, 21536368, 13032432, 18031054, 24678532, 15511076, 9099889, 8253323, 5219505, 1757169, 287496, 17424

Offset

1,5

Comments

For point x = (x_1,x_2,x_3,x_4) in the 4-dimensional lattice, we define the semisymmetric height of x as h_4'(x) := 3x_1 + x_2 - x_3 - 3x_4. The 4-dimensional balanced ballot path (multidimensional Dyck path) is a sequence of 4*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 >= x_4. T(n,k) is the number of 4-dimensional balanced ballot paths of 4*n steps such that the largest semisymmetric height reached by any point in the path is equal to k, i.e., for at least one intermediate point h_4'(x) = k, but for no intermediate points h_4'(x) > k. In this context, 1 <= k <= 4n.

It is important to note that, here, the semisymmetric height of a point in the 4-dimensional lattice is defined differently from the height defined in the comments of sequence A387987.

Links

Table of n, a(n) for n=1..66.

Ryota Inagaki and Dimana Pramatarova, On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers, arXiv:2604.04900 [math.CO], 2026. See p. 27.

Example

Triangle begins:
  1;
  1, 0, 1, 8, 4;
  1, 0, 3, 69, 48, 30, 151, 135, 25;
  1, 0, 7, 533, 553, 583, 4299, 5051, 1930, 4288, 4819, 1764, 196;
  ...

Prog

(Python)
from functools import lru_cache
@lru_cache(None)
def ht(t):
    return 3 * t[0] + t[1] - t[2] - 3*t[3]
def f(t, bd):
    count = 0
    if t == (0, 0, 0, 0):
        return 1
    if t[0] > 0 and t[0] - 1 >= t[1]:
        count += f((t[0] - 1, t[1], t[2], t[3]), bd)
    if t[1] > 0 and t[1] - 1 >= t[2] and ht((t[0], t[1] - 1, t[2], t[3])) <= bd:
        count += f((t[0], t[1] - 1, t[2], t[3]), bd)
    if t[2] > 0 and t[2] - 1 >= t[3] and ht((t[0], t[1], t[2] - 1, t[3])) <= bd:
        count += f((t[0], t[1], t[2] - 1, t[3]), bd)
    if t[3] > 0 and ht((t[0], t[1], t[2], t[3] - 1)) <= bd:
        count += f((t[0], t[1], t[2], t[3] -1), bd)
    return count
for n in range(1, 7):
    a = 0
    b = 0
    for k in range(4, 4*n+1):
        if k == 4:
            a = f((n, n, n, n), k)
            print(a)
        if k > 4:
            b = f((n, n, n, n), k)
            print(b-a)
            a = b

Crossrefs

Cf. A393571, A393573, A387987.

Row sums are A005790. Last entry of rows equal A001246. Last entry of rows equal the first column of A393571.

Sequence in context: A049255 A028577 A039662 * A228496 A259616 A247036

Adjacent sequences: A393591 A393592 A393593 * A393595 A393596 A393597

Keyword

nonn,tabf

Author

Ryota Inagaki and Dimana Pramatarova, Feb 23 2026

Status

approved