A288109 -id:A288109 - cp's OEIS frontend

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

Alois P. Heinz, Jun 05 2017

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.

			. 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;

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

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.

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);

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 *)

A287987 Number of Dyck paths of semilength n such that all positive levels have the same number of peaks.

Original entry on oeis.org

1, 1, 1, 3, 1, 8, 13, 13, 54, 132, 280, 547, 1219, 3904, 11107, 25082, 53777, 137751, 419831, 1257599, 3453557, 8911341, 22636845, 59890162, 172264224, 529706648, 1630328686, 4765347773, 13125989799, 35253234315, 97531470556, 287880507391, 894915519516

Offset: 0

Alois P. Heinz, Jun 03 2017

nonn

			. a(3) = 3:                         /\        /\
.                    /\/\/\      /\/  \      /  \/\  .
.
. a(5) = 8:
.                       /\/\      /\/\      /\/\
.      /\/\/\/\/\  /\/\/    \  /\/    \/\  /    \/\/\
.
.            /\        /\          /\        /\
.         /\/  \      /  \/\    /\/  \      /  \/\
.      /\/      \  /\/      \  /      \/\  /      \/\  .

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Counting lattice paths

Row sums of A288318.

Cf. A000108, A287845, A287846, A287993, A288109.

b:= proc(n, k, j) option remember; `if`(n=j, 1,
       add(binomial(i, k)*binomial(j-1, i-1-k)
         *b(n-j, k, i), i=1+k..min(j+k, n-j)))
    end:
a:= n-> 1+add(b(n, j$2), j=1..n/2):
seq(a(n), n=0..33);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Binomial[i, k]*Binomial[j - 1, i - 1 - k]*b[n - j, k, i], {i, 1 + k, Min[j + k, n - j]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n/2}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

cp's OEIS Frontend

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.

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