A393689 - OEIS

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.

%I #29 Mar 24 2026 09:43:24

%S 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,

%T 68,0,14,36,70,104,132,136,100,0,16,42,84,130,176,204,200,140,0,18,48,

%U 98,156,220,272,300,280,190,0,20,54,112,182,264,340,400,420,380,250

%N 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.

%C 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.

%H Paolo Xausa, <a href="/A393689/b393689.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).

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

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

%e Square array A(k, n) begins:

%e k\n | 2 3 4 5 6 7 8 9

%e -----+----------------------------------------

%e 1 0 2 6 14 26 44 68 100

%e 2 0 4 12 28 52 88 136 200

%e 3 0 6 18 42 78 132 204 300

%e 4 0 8 24 56 104 176 272 400

%e 5 0 10 30 70 130 220 340 500

%e 6 0 12 36 84 156 264 408 600

%e 7 0 14 42 98 182 308 476 700

%e 8 0 16 48 112 208 352 544 800

%e 9 0 18 54 126 234 396 612 900

%t A393689[n_, k_] := (k*(k - 2)*(2*k + 1) + If[OddQ[k], 3, 0])*n/12;

%t Table[A393689[n - k + 2, k], {n, 15}, {k, 2, n + 1}] (* _Paolo Xausa_, Mar 22 2026 *)

%o (Python)

%o def A(n, k):

%o if k % 2 == 0:

%o return k * (k - 2) * (2 * k + 1) * n // 12

%o else:

%o return (k * k - 1) * (2 * k - 3) * n // 12

%o def row_n(n, K):

%o return [A(n, k) for k in range(2, K + 1)]

%o for n in range(1, 10):

%o print(f"k={k}:", row_n(n, 9))

%Y Cf. A393571, A393573, A392248.

%K nonn,tabl,easy

%O 1,3

%A _Ryota Inagaki_ and _Dimana Pramatarova_, Feb 24 2026