A287987 -id:A287987 - cp's OEIS frontend

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

Alois P. Heinz, Jun 07 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(5,1) = 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;

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

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.

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);

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 *)

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.

A287846 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly one peak.

Original entry on oeis.org

1, 1, 0, 2, 0, 4, 6, 8, 24, 52, 96, 212, 504, 1072, 2352, 5288, 11928, 26800, 60336, 136304, 308928, 701248, 1593120, 3622016, 8245008, 18787360, 42836928, 97724384, 223052784, 509338816, 1163512032, 2658731648, 6077117376, 13893874624, 31771515648

Offset: 0

Alois P. Heinz, Jun 01 2017

nonn

All terms with n > 1 are even.

			. a(1) = 1:    /\  .
.
. a(3) = 2:     /\       /\
.            /\/  \     /  \/\  .
.
. a(5) = 4:
.                /\       /\         /\       /\
.             /\/  \     /  \/\   /\/  \     /  \/\
.          /\/      \ /\/      \ /      \/\ /      \/\ .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 15.
Wikipedia, Counting lattice paths

Column k=1 of A288318.

Cf. A000108, A281874, A287843, A287845, A287901, A287963, A287987, A289020.

b:= proc(n, j) option remember; `if`(n=j or n=0, 1, add(
       b(n-j, i)*binomial(j-1, i-2)*i, i=1..min(j+2, n-j)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);

b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*Binomial[j - 1, i - 2]*i, {i, 1, Min[j + 2, n - j]}]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 23 2018, translated from Maple *)

A287845 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly two peaks.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 6, 0, 9, 54, 138, 207, 360, 1368, 4545, 11304, 25182, 61605, 173916, 498798, 1347417, 3497328, 9147060, 24630669, 67414590, 184065966, 498495303, 1345622436, 3642036804, 9900361107, 26982011250, 73570082760, 200540053395, 546660151722

Offset: 0

Alois P. Heinz, Jun 01 2017

nonn

			. a(2) = 1:   /\/\  .
.
. a(5) = 3:
.
.               /\/\     /\/\     /\/\
.          /\/\/    \ /\/    \/\ /    \/\/\ .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 15.
Wikipedia, Counting lattice paths

Column k=2 of A288318.

Cf. A000108, A281874, A287843, A287846, A287963, A287987.

b:= proc(n, j) option remember;
      `if`(n=j or n=0, 1, add(b(n-j, i)*i*(i-1)/2
       *binomial(j-1, i-3), i=3..min(j+2, n-j)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..35);

b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*i*(i - 1)/2* Binomial[j - 1, i - 3], {i, 3, Min[j + 2, n - j]}]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

A287993 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.

Original entry on oeis.org

1, 1, 1, 1, 6, 10, 21, 52, 147, 564, 1651, 4440, 12499, 36853, 116476, 390774, 1352215, 4593736, 15057127, 48419013, 156073723, 511324062, 1713185811, 5878350249, 20574046540, 72771206715, 257475113013, 905430711156, 3160767910928, 10981916671027

Offset: 0

Alois P. Heinz, Jun 04 2017

nonn

			a(4) = 6:
                     /\      /\      /\          /\/\    /\/\
      /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\

Alois P. Heinz, Table of n, a(n) for n = 0..70
Wikipedia, Counting lattice paths

Cf. A000108, A287987, A287989.

b:= proc(n, s, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, s union {t}, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t={$max(1, i-j)..min(n-j, i-1)} minus s), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, {k}, k), k=1..n)):
seq(a(n), n=0..30);

b[n_, s_, j_] := b[n, s, j] = If[n==j, 1, Sum[Sum[b[n-j, s ~Union~ {t}, i]* Binomial[i, t]*Binomial[j-1, i-1-t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ s}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, {k}, k], {k, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 31 2018, from Maple *)

A288109 Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.

Original entry on oeis.org

1, 1, 2, 5, 9, 23, 56, 122, 323, 792, 2060, 5199, 13314, 35171, 94077, 249285, 662901, 1775244, 4806724, 13125887, 36107283, 99863241, 276784435, 768288783, 2143763275, 6037486060, 17171063218, 49187617277, 141512589597, 408293870713, 1181084207303

Offset: 0

Alois P. Heinz, Jun 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Counting lattice paths

Row sums of A288108.

Cf. A000108, A287987.

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:
a:= n-> 1 + add(b(n, j$2), j=1..n-1):
seq(a(n), n=0..33);

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]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n - 1}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 31 2018, from Maple *)

cp's OEIS Frontend

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.

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A287846 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly one peak.

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A287845 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly two peaks.

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A287993 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and all the level peak numbers are distinct.

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Maple

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A288109 Number of Dyck paths of semilength n such that all levels with peaks have exactly the same number of peaks.

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Author

Keywords

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Programs

Maple

Mathematica