A288318 Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 4, 3, 0, 0, 1, 0, 6, 6, 0, 0, 0, 1, 0, 8, 0, 4, 0, 0, 0, 1, 0, 24, 9, 20, 0, 0, 0, 0, 1, 0, 52, 54, 20, 5, 0, 0, 0, 0, 1, 0, 96, 138, 0, 45, 0, 0, 0, 0, 0, 1, 0, 212, 207, 16, 105, 6, 0, 0, 0, 0, 0, 1, 0, 504, 360, 200, 70, 84, 0, 0, 0, 0, 0, 0, 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

Formula

T(n,n) = 1.

T(n+1,n) = 0.

T(2*n+1,n) = (n+1) for n>0.

T(2*n+2,n) = A005564(n+1) for n>1.

T(3*n,n) = A000984(n) = binomial(2*n,n).

T(3*n+1,n) = 0.

T(3*n+2,n) = (n+1)^2 for n>0.

Example

. T(5,1) = 4:
.               /\        /\          /\        /\
.            /\/  \      /  \/\    /\/  \      /  \/\
.         /\/      \  /\/      \  /      \/\  /      \/\ .
.
. T(5,2) = 3:
.              /\/\      /\/\      /\/\
.         /\/\/    \  /\/    \/\  /    \/\/\  .
.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  2,  0,  1;
  0,  0,  0,  0, 1;
  0,  4,  3,  0, 0, 1;
  0,  6,  6,  0, 0, 0, 1;
  0,  8,  0,  4, 0, 0, 0, 1;
  0, 24,  9, 20, 0, 0, 0, 0, 1;
  0, 52, 54, 20, 5, 0, 0, 0, 0, 1;

Maple

b:= proc(n, k, j) option remember;
     `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
      *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
T:= (n, k)-> `if`(n=0, 1, 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[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
T[n_, k_] := If[n == 0, 1, 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, A287846, A287845, A288319, A288320, A288321, A288322, A288323, A288324, A288325, A288326.

Row sums give A287987.

Cf. A000108, A000984, A005564, A288108, A288940.

Sequence in context: A331671 A123758 A231642 * A354099 A219483 A239927

Adjacent sequences: A288315 A288316 A288317 * A288319 A288320 A288321

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 07 2017

Status

approved