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

0,8

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.

Links

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

Wikipedia, Counting lattice paths

Example

. 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;

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 *)

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.

Sequence in context: A356115 A363756 A130160 * A287822 A162169 A216954

Adjacent sequences: A288105 A288106 A288107 * A288109 A288110 A288111

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 05 2017

Status

approved