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%I #27 Oct 07 2018 18:13:23
%S 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,
%T 144,243,128,6,0,0,7,57,275,736,891,304,7,0,0,8,77,468,1750,3584,3159,
%U 704,8,0,0,9,100,735,3564,10625,16896,10935,1600,9,0
%N Number A(n,k) of ascending runs in {1,...,k}^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A229079/b229079.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = k^(n-1)*((n+1)*k+n-1)/2 for n>0, A(0,k) = 0.
%e A(4,1) = 4: [1,1,1,1].
%e 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].
%e 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].
%e A(1,4) = 4 = 1+1+1+1: [1], [2], [3], [4].
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 2, 7, 15, 26, 40, 57, 77, ...
%e 0, 3, 20, 63, 144, 275, 468, 735, ...
%e 0, 4, 52, 243, 736, 1750, 3564, 6517, ...
%e 0, 5, 128, 891, 3584, 10625, 25920, 55223, ...
%e 0, 6, 304, 3159, 16896, 62500, 182736, 453789, ...
%e 0, 7, 704, 10935, 77824, 359375, 1259712, 3647119, ...
%p A:= (n, k)-> `if`(n=0, 0, k^(n-1)*((n+1)*k+n-1)/2):
%p seq(seq(A(n,d-n), n=0..d), d=0..12);
%t 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 *)
%Y Columns k=0-10 give: A000004, A001477, A066373(n+1) for n>0, A229277, A229278, A229279, A229280, A229281, A229282, A229283, A229284.
%Y Rows n=0-10 give: A000004, A001477, A005449, A099721, A229146, A229147, A229148, A229149, A229150, A229151, A229152.
%Y Main diagonal gives A229078.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 14 2013
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