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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. 19
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = k^(n-1)*((n+1)*k+n-1)/2 for n>0, A(0,k) = 0. EXAMPLE 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, ... 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 *) 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. Sequence in context: A228275 A228250 A101164 * A254040 A062275 A138270 Adjacent sequences:  A229076 A229077 A229078 * A229080 A229081 A229082 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 14 2013 STATUS approved

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Last modified February 19 10:24 EST 2019. Contains 320310 sequences. (Running on oeis4.)