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

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