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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-Fran├žois 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: A228250 A341317 A101164 * A329331 A254040 A062275

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 January 20 08:44 EST 2022. Contains 350467 sequences. (Running on oeis4.)