A264909 Number A(n,k) of k-ascent sequences of length n with no consecutive repeated letters; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 6, 5, 0, 1, 1, 4, 12, 21, 16, 0, 1, 1, 5, 20, 54, 87, 61, 0, 1, 1, 6, 30, 110, 276, 413, 271, 0, 1, 1, 7, 42, 195, 670, 1574, 2213, 1372, 0, 1, 1, 8, 56, 315, 1380, 4470, 9916, 13205, 7795, 0

Offset: 0

Author

Alois P. Heinz, Nov 28 2015

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,   1,    1,    1,     1,     1,      1,      1, ...
  1,   1,    1,    1,     1,     1,      1,      1, ...
  0,   1,    2,    3,     4,     5,      6,      7, ...
  0,   2,    6,   12,    20,    30,     42,     56, ...
  0,   5,   21,   54,   110,   195,    315,    476, ...
  0,  16,   87,  276,   670,  1380,   2541,   4312, ...
  0,  61,  413, 1574,  4470, 10555,  21931,  41468, ...
  0, 271, 2213, 9916, 32440, 86815, 201761, 422128, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• S. Kitaev, J. Remmel, p-Ascent Sequences, arXiv:1503.00914 [math.CO], 2015.

Crossrefs

Columns k=1-10 give: A138265, A263852, A263853, A263854, A264910, A264911, A264912, A264913, A264914, A264915.

Rows k=0+1,2-4 give: A000012, A001477, A002378, A160378(n+1).

Main diagonal gives A264916.

Programs

Maple

b:= proc(n, k, i, t) option remember; `if`(n<1, 1, add(
      `if`(j=i, 0, b(n-1, k, j, t+`if`(j>i, 1, 0))), j=0..t+k))
    end:
A:= (n, k)-> b(n-1, k, 0$2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[n_, k_, i_, t_] := b[n, k, i, t] = If[n<1, 1, Sum[If[j == i, 0, b[n-1, k, j, t + If[j>i, 1, 0]]], {j, 0, t+k}]]; A[n_, k_] := b[n-1, k, 0, 0]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)