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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A113048 A123758 A231642 * A219483 A239927 A069846

Adjacent sequences:  A288315 A288316 A288317 * A288319 A288320 A288321

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 07 2017

STATUS

approved

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Last modified June 18 16:46 EDT 2019. Contains 324214 sequences. (Running on oeis4.)