A288108 Number T(n,k) of Dyck paths of semilength n such that each level has exactly k peaks or no peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 5, 2, 1, 1, 0, 13, 5, 3, 1, 1, 0, 31, 15, 4, 4, 1, 1, 0, 71, 27, 10, 7, 5, 1, 1, 0, 181, 76, 36, 11, 11, 6, 1, 1, 0, 447, 196, 83, 22, 19, 16, 7, 1, 1, 0, 1111, 548, 225, 81, 32, 31, 22, 8, 1, 1, 0, 2799, 1388, 573, 235, 60, 56, 48, 29, 9, 1, 1

Offset: 0

Author

Alois P. Heinz, Jun 05 2017

Keywords

• nonn
• tabl

Comments

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(0,k) = 1 and T(n,k) = 0 if k > n > 0.

Examples

			. T(5,2) = 5:                                        /\/\
.                                       /\  /\      /    \
.      /\/\      /\/\      /\/\        /  \/  \    /      \
. /\/\/    \  /\/    \/\  /    \/\/\  /        \  /        \ .
.
. T(5,3) = 3:
.                                       /\/\/\
.              /\  /\/\    /\/\  /\    /      \
.             /  \/    \  /    \/  \  /        \ .
.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,  1;
  0,   3,  1,  1;
  0,   5,  2,  1,  1;
  0,  13,  5,  3,  1,  1;
  0,  31, 15,  4,  4,  1, 1;
  0,  71, 27, 10,  7,  5, 1, 1;
  0, 181, 76, 36, 11, 11, 6, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Counting lattice paths

Crossrefs

Columns k=0-10 give: A000007, A281874, A287843, A288110, A288111, A288112, A288113, A288114, A288115, A288116, A288117.

Row sums give A288109.

T(2n,n) gives A156043.

Cf. A000108, A288318.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
      b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
       *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
T:= (n, k)-> b(n, k$2):
seq(seq(T(n, k), k=0..n), n=0..14);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[j - 1, i - 1] + Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
T[n_, k_] := b[n, k, k];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)