A134481 -id:A134481 - cp's OEIS frontend

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

Joerg Arndt and Alois P. Heinz, May 05 2014

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

			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.

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

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).

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);

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 *)

A360546 Triangle read by rows: T(n, m) = (n+1-m)C(2n+2-m, m)C(3n-3m+2, n-m+1)/(2n-m+2).

Original entry on oeis.org

1, 5, 2, 28, 20, 3, 165, 168, 50, 4, 1001, 1320, 588, 100, 5, 6188, 10010, 5940, 1568, 175, 6, 38760, 74256, 55055, 19800, 3528, 280, 7, 245157, 542640, 482664, 220220, 54450, 7056, 420, 8, 1562275, 3922512, 4069800, 2252432, 715715, 130680, 12936, 600, 9

Offset: 0

Vladimir Kruchinin, Feb 11 2023

			Triangle begins:
     1;
     5,     2;
    28,    20,    3;
   165,   168,   50,    4;
  1001,  1320,  588,  100,   5;
  6188, 10010, 5940, 1568, 175, 6;

Cf. A025174, A134481.

A360546 := proc(n, k) m := n-k+1; (1/3)*binomial(3*m, m)*binomial(m + n, k) end:
seq(print(seq(A360546(n, k), k = 0..n)), n = 0..8); # Peter Luschny, Feb 11 2023

Maxima
```
T(n,m):=if n
				
```

G.f.: -1/(2*x) + (sqrt(3)*cot((1/3)*arcsin((3*sqrt(3)*sqrt(x))/(2- 2*x*y))))/ (2*sqrt(x*(-27*x + 4*(-1+x*y)^2))).

A134480 A134478 * A000012.

Original entry on oeis.org

1, 3, 2, 9, 7, 4, 18, 15, 11, 6, 30, 26, 21, 15, 8, 45, 40, 34, 27, 19, 10, 63, 57, 50, 42, 33, 23, 12, 84, 77, 69, 60, 50, 39, 27, 14, 108, 100, 91, 81, 70, 58, 45, 31, 16, 135, 126, 116, 105, 93, 80, 66, 51, 35, 18

Offset: 0

Gary W. Adamson, Oct 27 2007

Row sums = A134481: (1, 5, 20, 50, 100, 175, ...).

Left border = A134479.

			First few rows of the triangle:
   1;
   3,  2;
   9,  7,  4;
  18, 15, 11,  6;
  30, 26, 21, 15,  8;
  45, 40, 34, 27, 19, 10;
  63, 57, 50, 42, 33, 23, 12;
  ...

Cf. A134478, A134481, A134479.

A134478 * A000012 as infinite lower triangular matrices. Triangle read by rows, partial sums of A134478 terms starting from the right.

cp's OEIS Frontend

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.

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A360546 Triangle read by rows: T(n, m) = (n+1-m)C(2n+2-m, m)C(3n-3m+2, n-m+1)/(2n-m+2).

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A134480 A134478 * A000012.

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cp's OEIS Frontend

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.

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Maple

Mathematica

A360546 Triangle read by rows: T(n, m) = (n+1-m)*C(2*n+2-m, m)*C(3*n-3*m+2, n-m+1)/(2*n-m+2).

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Maple

Maxima

Formula

A134480 A134478 * A000012.

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Formula

A360546 Triangle read by rows: T(n, m) = (n+1-m)C(2n+2-m, m)C(3n-3m+2, n-m+1)/(2n-m+2).