A242153 Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 16, 20, 12, 4, 1, 0, 61, 80, 50, 20, 5, 1, 0, 271, 366, 240, 100, 30, 6, 1, 0, 1372, 1897, 1281, 560, 175, 42, 7, 1, 0, 7795, 10976, 7588, 3416, 1120, 280, 56, 8, 1, 0, 49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0

Offset

0,7

Comments

In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

Links

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Example

Triangle T(n,k) begins:
00:      1;
01:      1,     0;
02:      1,     1,     0;
03:      2,     2,     1,     0;
04:      5,     6,     3,     1,    0;
05:     16,    20,    12,     4,    1,    0;
06:     61,    80,    50,    20,    5,    1,   0;
07:    271,   366,   240,   100,   30,    6,   1,  0;
08:   1372,  1897,  1281,   560,  175,   42,   7,  1, 0;
09:   7795, 10976,  7588,  3416, 1120,  280,  56,  8, 1, 0;
10:  49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0;
...
The 15 ascent sequences of length 4 (dots denote zeros) with their number of flat steps are:
01:  [ . . . . ]   3
02:  [ . . . 1 ]   2
03:  [ . . 1 . ]   1
04:  [ . . 1 1 ]   2
05:  [ . . 1 2 ]   1
06:  [ . 1 . . ]   1
07:  [ . 1 . 1 ]   0
08:  [ . 1 . 2 ]   0
09:  [ . 1 1 . ]   1
10:  [ . 1 1 1 ]   2
11:  [ . 1 1 2 ]   1
12:  [ . 1 2 . ]   0
13:  [ . 1 2 1 ]   0
14:  [ . 1 2 2 ]   1
15:  [ . 1 2 3 ]   0
There are 5 sequences without flat steps, 6 with one flat step, etc., giving row [5, 6, 3, 1, 0] for n=4.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
      `if`(j=i, x, 1) *b(n-1, j, t+`if`(j>i, 1, 0)), j=0..t+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, -1$2)):
seq(T(n), n=0..12);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == i, x, 1]*b[n-1, j, t + If[j>i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][ b[n, -1, -1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A138265, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163.

Row sums give A022493.

T(2n,n) gives A242164.

Main diagonal and lower diagonals give: A000007, A000012, A000027(n+1), A002378(n+1), A134481(n+1), A130810(n+4).

Cf. A137251 (the same for ascents), A238858 (the same for descents).

Sequence in context: A120568 A321960 A189233 * A065066 A266291 A064045

Adjacent sequences: A242150 A242151 A242152 * A242154 A242155 A242156

Keyword

nonn,tabl

Author

Joerg Arndt and Alois P. Heinz, May 05 2014

Status

approved