A229079 Number A(n,k) of ascending runs in {1,...,k}^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 15, 20, 4, 0, 0, 5, 26, 63, 52, 5, 0, 0, 6, 40, 144, 243, 128, 6, 0, 0, 7, 57, 275, 736, 891, 304, 7, 0, 0, 8, 77, 468, 1750, 3584, 3159, 704, 8, 0, 0, 9, 100, 735, 3564, 10625, 16896, 10935, 1600, 9, 0

Offset: 0

Author

Alois P. Heinz, Sep 14 2013

Keywords

• nonn
• tabl

Examples

			A(4,1) = 4: [1,1,1,1].
A(3,2) = 20 = 3+3+2+3+2+2+2+3: [1,1,1], [2,1,1], [1,2,1], [2,2,1], [1,1,2], [2,1,2], [1,2,2], [2,2,2].
A(2,3) = 15 = 2+2+2+1+2+2+1+1+2: [1,1], [2,1], [3,1], [1,2], [2,2], [3,2], [1,3], [2,3], [3,3].
A(1,4) = 4 = 1+1+1+1: [1], [2], [3], [4].
Square array A(n,k) begins:
  0, 0,   0,     0,     0,      0,       0,       0, ...
  0, 1,   2,     3,     4,      5,       6,       7, ...
  0, 2,   7,    15,    26,     40,      57,      77, ...
  0, 3,  20,    63,   144,    275,     468,     735, ...
  0, 4,  52,   243,   736,   1750,    3564,    6517, ...
  0, 5, 128,   891,  3584,  10625,   25920,   55223, ...
  0, 6, 304,  3159, 16896,  62500,  182736,  453789, ...
  0, 7, 704, 10935, 77824, 359375, 1259712, 3647119, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000004, A001477, A066373(n+1) for n>0, A229277, A229278, A229279, A229280, A229281, A229282, A229283, A229284.

Rows n=0-10 give: A000004, A001477, A005449, A099721, A229146, A229147, A229148, A229149, A229150, A229151, A229152.

Main diagonal gives A229078.

Programs

Maple

A:= (n, k)-> `if`(n=0, 0, k^(n-1)*((n+1)*k+n-1)/2):
seq(seq(A(n,d-n), n=0..d), d=0..12);

Mathematica

a[, 0] = a[0, ] = 0; a[n_, k_] := k^(n-1)*((n+1)*k+n-1)/2; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 09 2013 *)

Formula

A(n,k) = k^(n-1)*((n+1)*k+n-1)/2 for n>0, A(0,k) = 0.