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

0,13

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

S. Kitaev, J. Remmel, p-Ascent Sequences, arXiv:1503.00914 [math.CO], 2015.

Example

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, ...

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 *)

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.

Sequence in context: A112555 A108561 A174626 * A104579 A079531 A182882

Adjacent sequences: A264906 A264907 A264908 * A264910 A264911 A264912

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 28 2015

Status

approved