A393689 Array read by antidiagonals: A(n, k) is the minimum number of whole unit squares under the polygonal path (i, h_k(v_i)), associated to a k-dimensional balanced ballot path, n >= 1, k >= 2.

0, 0, 2, 0, 4, 6, 0, 6, 12, 14, 0, 8, 18, 28, 26, 0, 10, 24, 42, 52, 44, 0, 12, 30, 56, 78, 88, 68, 0, 14, 36, 70, 104, 132, 136, 100, 0, 16, 42, 84, 130, 176, 204, 200, 140, 0, 18, 48, 98, 156, 220, 272, 300, 280, 190, 0, 20, 54, 112, 182, 264, 340, 400, 420, 380, 250

Offset

1,3

Comments

For a point x = (x_1, x_2, ..., x_k) in the k-dimensional lattice, we define the semisymmetric height of x as h_k'(x) = Sum_{i=1..k} (k+1-2*i)*x_i. A k-dimensional balanced ballot path (also called a multidimensional Dyck path), is a sequence of k*n steps starting at (0, ..., 0) and ending at (n, ..., n), such that each step is a standard unit vector and each point of the path satisfies x_1 >= x_2 >= ... >= x_k. A(n, k) is defined as the minimum number of whole unit squares under the path (0, 0), (1, h_k'(v_1)), ..., (k*n, h_k'(v_{k*n})) (the 2-dimensional image under the height map), where 0, v_1, v_2, ..., v_{k*n} are the intermediate points of a k-dimensional balanced ballot path.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Formula

A(n, k) = k*(k-2)*(2*k+1)*n/12 for even k and

A(n, k) = (k^2-1)*(2*k-3)*n/12 for odd k.

Example

Square array A(k, n) begins:
k\n  |   2    3    4    5    6    7    8     9
-----+----------------------------------------
  1      0    2    6   14   26   44   68   100
  2      0    4   12   28   52   88  136   200
  3      0    6   18   42   78  132  204   300
  4      0    8   24   56  104  176  272   400
  5      0   10   30   70  130  220  340   500
  6      0   12   36   84  156  264  408   600
  7      0   14   42   98  182  308  476   700
  8      0   16   48  112  208  352  544   800
  9      0   18   54  126  234  396  612   900

Mathematica

A393689[n_, k_] := (k*(k - 2)*(2*k + 1) + If[OddQ[k], 3, 0])*n/12;
Table[A393689[n - k + 2, k], {n, 15}, {k, 2, n + 1}] (* Paolo Xausa, Mar 22 2026 *)

Prog

(Python)
def A(n, k):
    if k % 2 == 0:
        return k * (k - 2) * (2 * k + 1) * n // 12
    else:
        return (k * k - 1) * (2 * k - 3) * n // 12
def row_n(n, K):
    return [A(n, k) for k in range(2, K + 1)]
for n in range(1, 10):
    print(f"k={k}:", row_n(n, 9))

Crossrefs

Cf. A393571, A393573, A392248.

Sequence in context: A096984 A213723 A104601 * A233673 A319931 A192133

Adjacent sequences: A393686 A393687 A393688 * A393690 A393691 A393692

Keyword

nonn,tabl,easy

Author

Ryota Inagaki and Dimana Pramatarova, Feb 24 2026

Status

approved