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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.)