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
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
Programs
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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);
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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.