A008930 -id:A008930 - cp's OEIS frontend

A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.

1, 1, 2, 4, 7, 14, 26, 51, 99, 195, 383, 759, 1504, 2988, 5944, 11840, 23602, 47084, 93975, 187647, 374812, 748857, 1496487, 2991017, 5978900, 11952780, 23897506, 47782081, 95543378, 191053334, 382052880, 764019152, 1527898772, 3055572646, 6110782652, 12220980359

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

			There are a(6) = 26 such compositions of 6:
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 2 1 ]
04:  [ 1 1 1 3 ]
05:  [ 1 1 2 1 1 ]
06:  [ 1 1 2 2 ]
07:  [ 1 1 3 1 ]
08:  [ 1 2 1 1 1 ]
09:  [ 1 2 1 2 ]
10:  [ 1 2 2 1 ]
11:  [ 1 2 3 ]
12:  [ 2 1 1 1 1 ]
13:  [ 2 1 1 2 ]
14:  [ 2 1 2 1 ]
15:  [ 2 1 3 ]
16:  [ 2 2 1 1 ]
17:  [ 2 2 2 ]
18:  [ 2 3 1 ]
19:  [ 3 1 1 1 ]
20:  [ 3 1 2 ]
21:  [ 3 2 1 ]
22:  [ 3 3 ]
23:  [ 4 1 1 ]
24:  [ 4 2 ]
25:  [ 5 1 ]
26:  [ 6 ]

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-j, j+1), j=1..n),
       add(b(n-j, i+1), j=1..min(n,i))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 25 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, 1, Min[n, i]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(n) ~ c * 2^n, where c = 1/2 - QPochhammer(1/2)/2 = 0.3556059524566987893605501390353846099555440475796571079426294669... - Vaclav Kotesovec, May 01 2014, updated Mar 17 2024

A238860 Partitions with superdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 >= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 23, 26, 35, 43, 53, 64, 79, 91, 113, 135, 166, 197, 237, 277, 331, 387, 459, 541, 646, 754, 888, 1032, 1204, 1395, 1626, 1882, 2196, 2542, 2952, 3404, 3934, 4507, 5182, 5935, 6812, 7800, 8947, 10225, 11709, 13354, 15231, 17314, 19685, 22316, 25323, 28686, 32524, 36817, 41695

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

			There are a(13) = 23 such partitions of 13:
01:  [ 1 2 3 7 ]
02:  [ 1 2 4 6 ]
03:  [ 1 2 5 5 ]
04:  [ 1 2 10 ]
05:  [ 1 3 3 6 ]
06:  [ 1 3 4 5 ]
07:  [ 1 3 9 ]
08:  [ 1 4 4 4 ]
09:  [ 1 4 8 ]
10:  [ 1 5 7 ]
11:  [ 1 6 6 ]
12:  [ 1 12 ]
13:  [ 2 3 8 ]
14:  [ 2 4 7 ]
15:  [ 2 5 6 ]
16:  [ 2 11 ]
17:  [ 3 4 6 ]
18:  [ 3 5 5 ]
19:  [ 3 10 ]
20:  [ 4 9 ]
21:  [ 5 8 ]
22:  [ 6 7 ]
23:  [ 13 ]

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 18, 24, 32, 41, 55, 72, 95, 125, 164, 212, 275, 355, 459, 592, 763, 980, 1257, 1605, 2044, 2598, 3298, 4179, 5290, 6685, 8435, 10623, 13353, 16751, 20978, 26228, 32746, 40831, 50850, 63247, 78569, 97475, 120770, 149429, 184641, 227853, 280832, 345722, 425134, 522232, 640847, 785604

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

			There are a(12) = 24 such compositions of 12:
01:  [ 1 2 3 6 ]
02:  [ 1 2 4 5 ]
03:  [ 1 2 5 4 ]
04:  [ 1 2 9 ]
05:  [ 1 3 3 5 ]
06:  [ 1 3 4 4 ]
07:  [ 1 3 8 ]
08:  [ 1 4 3 4 ]
09:  [ 1 4 7 ]
10:  [ 1 5 6 ]
11:  [ 1 6 5 ]
12:  [ 1 7 4 ]
13:  [ 1 8 3 ]
14:  [ 1 11 ]
15:  [ 2 3 7 ]
16:  [ 2 4 6 ]
17:  [ 2 5 5 ]
18:  [ 2 6 4 ]
19:  [ 2 10 ]
20:  [ 3 4 5 ]
21:  [ 3 9 ]
22:  [ 4 8 ]
23:  [ 5 7 ]
24:  [ 12 ]

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Cf. A238860 (partitions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-j, j+1), j=1..n),
       add(b(n-j, i+1), j=i..n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, i, n}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf = 1 + sum(n=1, N, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) );
v=Vec(gf) \\ Joerg Arndt, Mar 30 2014

G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [Joerg Arndt, Mar 30 2014]

A238876 Partitions with subdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 <= i.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 24, 34, 46, 58, 76, 97, 126, 166, 209, 262, 333, 422, 529, 667, 833, 1024, 1268, 1567, 1934, 2385, 2911, 3549, 4319, 5237, 6340, 7675, 9274, 11164, 13404, 16046, 19173, 22889, 27278, 32458, 38574, 45750, 54140, 63976, 75449, 88848, 104503, 122773, 144077, 168860, 197609, 230916, 269494

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

The number of such partitions that start with part p0 = 1 are given in A238875.

			The a(9) = 20 such partitions are:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 3 ]
04:  [ 1 1 1 1 1 2 2 ]
05:  [ 1 1 1 1 1 4 ]
06:  [ 1 1 1 1 2 3 ]
07:  [ 1 1 1 1 5 ]
08:  [ 1 1 1 2 2 2 ]
09:  [ 1 1 1 2 4 ]
10:  [ 1 1 1 3 3 ]
11:  [ 1 1 2 2 3 ]
12:  [ 1 1 3 4 ]
13:  [ 1 2 2 2 2 ]
14:  [ 1 2 2 4 ]
15:  [ 1 2 3 3 ]
16:  [ 2 2 2 3 ]
17:  [ 2 3 4 ]
18:  [ 3 3 3 ]
19:  [ 4 5 ]
20:  [ 9 ]

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A238859 (compositions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

A177517 Triangle T(n,k) read by rows defined by recurrence T(n,1)=A000007(n-1) and T(n,k) = sum_{i=1..k-1} T(n-i,k-1) if k>1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 1, 5, 4, 1, 0, 0, 0, 0, 6, 9, 5, 1, 0, 0, 0, 0, 5, 15, 14, 6, 1, 0, 0, 0, 0, 3, 20, 29, 20, 7, 1, 0, 0, 0, 0, 1, 22, 49, 49, 27, 8, 1, 0, 0, 0, 0, 0, 20, 71, 98, 76, 35, 9, 1, 0, 0, 0, 0, 0, 15, 90, 169, 174, 111, 44, 10, 1, 0, 0, 0, 0, 0, 9, 101, 259, 343, 285, 155, 54, 11, 1, 0, 0, 0, 0, 0, 4, 101, 359, 602, 628, 440, 209, 65, 12, 1, 0, 0, 0, 0, 0, 1, 90, 455, 961, 1230, 1068, 649, 274, 77, 13, 1

Offset: 1

Mats Granvik, Roger L. Bagula, Dec 11 2010

A008302 is the main entry for this triangle.
Essentially A060701 which is equal to this table beginning from the second column.
The recurrence formula is similar to the recurrence for A177978.

			1,
0,1,
0,0,1,
0,0,1,1,
0,0,0,2,1,
0,0,0,2,3,1,
0,0,0,1,5,4,1,
0,0,0,0,6,9,5,1,
0,0,0,0,5,15,14,6,1,
0,0,0,0,3,20,29,20,7,1,
0,0,0,0,1,22,49,49,27,8,1

Alois P. Heinz, Table of n, a(n) for n = 1..20301

Cf. A008302, A060701, A177978, A175105. Column sums are A000142. Row sums are A008930.

t[1, 1] = 1; t[n_, 1] = 0; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0];
Flatten[Table[t[n, k], {n, 12}, {k, n}]]
(* Robert G. Wilson v, Jun 24 2011 *) (* corrected by Mats Granvik, Jan 23 2012 *)

T(n,k) = A008302(k-2,n-k), n>=k>1. - R. J. Mathar, Dec 15 2010

A351983 Number of integer compositions of n with exactly one part above the diagonal.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 18, 35, 67, 131, 257, 505, 996, 1973, 3915, 7781, 15486, 30855, 61527, 122764, 245069, 489412, 977673, 1953515, 3904108, 7803545, 15599618, 31187269, 62355347, 124679883, 249310255, 498540890, 996953659, 1993701032, 3987069747, 7973603891

Offset: 0

Gus Wiseman, Apr 02 2022

nonn

			The a(2) = 1 through a(6) = 18 compositions:
  (2)  (3)   (4)    (5)     (6)
       (21)  (13)   (14)    (15)
             (22)   (32)    (42)
             (31)   (41)    (51)
             (211)  (131)   (114)
                    (212)   (132)
                    (221)   (141)
                    (311)   (213)
                    (2111)  (222)
                            (312)
                            (321)
                            (411)
                            (1311)
                            (2112)
                            (2121)
                            (2211)
                            (3111)
                            (21111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for permutations is A000295, weak A057427.
The version for partitions is A002620, weak A001477.
The weak version is A177510.
The version for fixed points is A240736, nonfixed A352520.
This is column k = 1 of A352524; column k = 0 is A008930.
A238349 counts compositions by fixed points, first column A238351.
A352521 counts compositions by strong nonexcedances, first column A219282.
A352522 counts compositions by weak nonexcedances, first column A238874.
A352523 counts compositions by nonfixed points, first column A010054.
A352524 counts compositions by strong excedances, first column A008930.
A352525 counts compositions by weak excedances, first column A177510.
Cf. A088218, A098825, A115994, A238352, A330644, A352516.

pless[y_]:=Length[Select[Range[Length[y]],#

S(v,u,c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=0); v[1]=1; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1)), O(x^2)); s+=apply(p->polcoef(p,1), v)); s} \\ Andrew Howroyd, Jan 02 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

A288140 Number of Dyck paths of semilength n such that the number of peaks is weakly decreasing from lower to higher levels.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 28, 63, 177, 455, 1233, 3383, 9359, 26809, 77078, 223201, 653982, 1934508, 5783712, 17431660, 52879184, 161386859, 495432345, 1530191918, 4754079840, 14849407892, 46604383972, 146897291083, 464892421363, 1477052536749, 4711124635655

Offset: 0

Alois P. Heinz, Jun 05 2017

nonn

			. a(5) = 12:
.                       /\         /\         /\
.   /\/\/\/\/\   /\/\/\/  \   /\/\/  \/\   /\/  \/\/\
.
.    /\               /\/\       /\/\       /\/\
.   /  \/\/\/\   /\/\/    \   /\/    \/\   /    \/\/\
.
.         /\         /\           /\         /\
.      /\/  \       /  \/\     /\/  \       /  \/\
.   /\/      \   /\/      \   /      \/\   /      \/\  .

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Counting lattice paths

Cf. A000108, A008930, A048285, A288141, A288146, A288147.

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t=max(k, i-j)..min(n-j, i-1)), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..31);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[k, i - j], Min[n - j, i - 1]}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 29 2018, from Maple *)

A288141 Number of Dyck paths of semilength n such that the number of peaks is strongly decreasing from lower to higher levels.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 10, 22, 46, 148, 324, 722, 1843, 4634, 12537, 34248, 95711, 266761, 724689, 1983267, 5553902, 15900083, 46201546, 135511171, 400668869, 1189723253, 3535186203, 10516298421, 31405658622, 94378367065, 285623516777, 870481565252, 2671088133010

Offset: 0

Alois P. Heinz, Jun 05 2017

nonn

			a(5) = 5:
                     /\        /\        /\        /\
  /\/\/\/\/\  /\/\/\/  \  /\/\/  \/\  /\/  \/\/\  /  \/\/\/\

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Counting lattice paths

Cf. A000108, A008930, A048285, A288140, A288146, A288147.

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t=max(k+1, i-j)..min(n-j, i-1)), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..34);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[k + 1, i - j], Min[n - j, i - 1]}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 29 2018, from Maple *)

A288146 Number of Dyck paths of semilength n such that the number of peaks is weakly increasing from lower to higher levels and no positive level is peakless.

Original entry on oeis.org

1, 1, 1, 3, 3, 13, 28, 65, 199, 540, 1468, 4188, 12328, 36870, 110181, 331226, 1012241, 3137822, 9796834, 30695164, 96658857, 306575170, 979485119, 3148413910, 10169223709, 32983822120, 107413795300, 351235602807, 1153308804255, 3802294411213, 12581993628872

Offset: 0

Alois P. Heinz, Jun 05 2017

nonn

			a(3) = 3:
                           /\       /\
             /\/\/\     /\/  \     /  \/\
a(4) = 3:
                          /\/\     /\/\
            /\/\/\/\   /\/    \   /    \/\

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Counting lattice paths

Cf. A000108, A008930, A048285, A288140, A288141, A288147.

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t=max(1, i-j)..min(k, n-j, i-1)), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..34);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[1, i - j], Min[k, n - j, i - 1]}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 29 2018, from Maple *)

A288147 Number of Dyck paths of semilength n such that the number of peaks is strongly increasing from lower to higher levels and no positive level is peakless.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 12, 31, 68, 186, 506, 1299, 3481, 9712, 27692, 79587, 232743, 694896, 2086245, 6248158, 18771510, 57007483, 175149700, 542313513, 1688360997, 5288335561, 16679137617, 52933231538, 168768966207, 539981776609, 1733555552587, 5587076558809

Offset: 0

Alois P. Heinz, Jun 05 2017

nonn

			a(5) = 6:
                     /\  /\      /\    /\
    /\/\/\/\/\    /\/  \/  \    /  \/\/  \
.
     /\  /\          /\/\/\      /\/\/\
    /  \/  \/\    /\/      \    /      \/\

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Counting lattice paths

Cf. A000108, A008930, A048285, A288140, A288141, A288146.

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t=max(1, i-j)..min(k-1, n-j, i-1)), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..34);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[1, i - j], Min[k - 1, n - j, i - 1]}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 29 2018, from Maple *)

cp's OEIS Frontend

A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.

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A238860 Partitions with superdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 >= i.

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A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.

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A238876 Partitions with subdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 <= i.

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A177517 Triangle T(n,k) read by rows defined by recurrence T(n,1)=A000007(n-1) and T(n,k) = sum_{i=1..k-1} T(n-i,k-1) if k>1.

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A351983 Number of integer compositions of n with exactly one part above the diagonal.

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Extensions

A288140 Number of Dyck paths of semilength n such that the number of peaks is weakly decreasing from lower to higher levels.

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A288141 Number of Dyck paths of semilength n such that the number of peaks is strongly decreasing from lower to higher levels.

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