A392248 Array read by antidiagonals: A(n, k) is the maximum number of whole unit squares under the polygonal path (i, h_k'(v_i)), associated with a k-dimensional balanced ballot path, n >= 1, k >= 2.

0, 2, 2, 6, 8, 6, 12, 18, 20, 14, 20, 32, 42, 40, 26, 30, 50, 72, 78, 70, 44, 42, 72, 110, 128, 132, 112, 68, 56, 98, 156, 190, 212, 204, 168, 100, 72, 128, 210, 264, 310, 320, 300, 240, 140, 90, 162, 272, 350, 426, 460, 464, 420, 330, 190

Offset

1,2

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 maximum 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).

Peter Luschny, Illustrating the Array.

Formula

Let m = floor(k/2), then

A(k, n) = m^2*n*(n-1) + n*(m-1)*m*(4*m+1)/3 for even k and

A(k, n) = (m+1)*m*n*(n-1) + n*(m+1)*m*(4*m-1)/3 for odd k.

From Peter Luschny, Jan 09 2026: (Start)

Arow(k) = n -> floor((k/2)^2)*n^2 + n*(k^3 - 3*k^2 - k + 3*(k mod 2))/6.

Arow(k) = n -> A391994(n)*n + A002620(k)*n^2.

gf(k) = (2*A391995(k)*x^2 - A212964(k)*x)/(x^3 - 3*x^2 + 3*x - 1) is the g.f. of row k. (End)

Example

Square array A(k, n) begins:
  k\n |  1    2    3    4    5    6    7    8    9
  ----+--------------------------------------------
   2  |  0    2    6   12   20   30   42   56   72 ...
   3  |  2    8   18   32   50   72   98  128  162 ...
   4  |  6   20   42   72  110  156  210  272  342 ...
   5  | 14   40   78  128  190  264  350  448  558 ...
   6  | 26   70  132  212  310  426  560  712  882 ...
   7  | 44  112  204  320  460  624  812 1024 1260 ...
   8  | 68  168  300  464  660  888 1148 1440 1764 ...
   9  |100  240  420  640  900 1200 1540 1920 2340 ...
  ...
  A(2, n) =    n^2 -    n;  A002378
  A(3, n) =  2*n^2       ;  A001105
  A(4, n) =  4*n^2 +  2*n;  A002943
  A(5, n) =  6*n^2 +  8*n;  A391993
  A(6, n) =  9*n^2 + 17*n;
  A(7, n) = 12*n^2 + 32*n;
  A(8, n) = 16*n^2 + 52*n;
  A(9, n) = 20*n^2 + 80*n;  A208375 (empirical)

Maple

A := proc(k, n) m := iquo(k, 2); if irem(k, 2) = 0 then
    m*m*n*(n-1) + n*(m-1)*m*(4*m+1)/3 else
(m+1)*m*n*(n-1) + n*(m+1)*m*(4*m-1)/3 fi end:
for k from 2 to 9 do lprint([k], seq(A(k, n), n = 1..12)) od;
# Alternative: as a list of rows:
Arow := k -> n -> floor((k/2)^2)*n^2 + n*(k^3 - 3*k^2 - k + 3*modp(k, 2))/6:
seq(lprint(seq(Arow(k)(n), n = 1..9)), k = 2..9);
# Alternative: by the generating functions of the rows:
A391995 := n -> (4*n^3 - 18*n^2 - 4*n + 9*(1-(-1)^n))/48:
A212964 := n -> (4*n^3 -  6*n^2 - 4*n + 3*(1-(-1)^n))/24:
ArowByGf := proc(k, len) local gf, ser, n;
gf := (2*A391995(k)*x^2 - A212964(k)*x)/(x^3 - 3*x^2 + 3*x - 1);
ser := series(gf, x, len + 1): seq(coeff(ser, x, n), n = 1..len) end:
for k from 2 to 9 do ArowByGf(k, 9) od;
# Peter Luschny, Jan 08 2026

Mathematica

A392248[k_, n_] := Quotient[k^2, 4]*n^2 + n*(3*Mod[k, 2] + k*((k - 3)*k - 1))/6;
Table[A392248[n + 1, k - n], {k, 2, 11}, {n, k - 1}] (* Paolo Xausa, Jan 13 2026 *)

Prog

(Python)
def A(k, n):
    m = k // 2
    if k % 2 == 0:
        return m*m*n*(n-1) + n*(m-1)*m*(4*m+1) // 3
    else:
        return m*(m+1)*n*(n-1) + (4*m-1)*m*(m+1)*n // 3
def row_k(k, N): return [A(k, n) for n in range(1, N+1)]
for k in range(1, 14): print(f"k={k}:", row_k(k, 14))

Crossrefs

Cf. A000108, A057571.

Cf. A002378, A001105, A001105, A002620, A208375, A212964, A391993, A391994, A391995.

Sequence in context: A306688 A092522 A116542 * A231131 A142243 A269722

Adjacent sequences: A392245 A392246 A392247 * A392249 A392250 A392251

Keyword

nonn,tabl

Author

Ryota Inagaki and Dimana Pramatarova, Jan 04 2026

Status

approved