A008930 -id:A008930 - cp's OEIS frontend

A352521 Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).

1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 0, 4, 5, 3, 3, 1, 0, 6, 8, 7, 6, 4, 1, 0, 9, 12, 15, 12, 10, 5, 1, 0, 13, 19, 27, 25, 22, 15, 6, 1, 0, 18, 32, 43, 51, 46, 37, 21, 7, 1, 0, 25, 51, 70, 94, 94, 83, 58, 28, 8, 1, 0, 35, 77, 117, 162, 184, 176, 141, 86, 36, 9, 1, 0

Offset: 0

Gus Wiseman, Mar 22 2022

			Triangle begins:
   1
   1   0
   1   1   0
   2   1   1   0
   3   2   2   1   0
   4   5   3   3   1   0
   6   8   7   6   4   1   0
   9  12  15  12  10   5   1   0
  13  19  27  25  22  15   6   1   0
  18  32  43  51  46  37  21   7   1   0
  25  51  70  94  94  83  58  28   8   1   0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
  (6)    (51)   (312)   (1113)   (11112)  (111111)  .
  (15)   (114)  (411)   (1122)   (11121)
  (24)   (132)  (1131)  (2112)   (11211)
  (33)   (141)  (1212)  (2121)   (21111)
  (42)   (213)  (1221)  (3111)
  (123)  (222)  (1311)  (12111)
         (231)  (2211)
         (321)

Andrew Howroyd, Table of n, a(n) for n = 0..1325
MathOverflow, Why 'excedances' of permutations? [closed].

Row sums are A011782.
The version for partitions is A114088.
Row sums without the last term are A131577.
The version for permutations is A173018.
Column k = 0 is A219282.
The corresponding rank statistic is A352514.
The weak version is A352522, first column A238874, rank statistic A352515.
The opposite version is A352524, first column A008930, rank stat A352516.
The weak opposite version is A352525, first col A177510, rank stat A352517.
A008292 is the triangle of Eulerian numbers (version without zeros).
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352490 is the strong nonexcedance set of A122111.
A352523 counts comps by nonfixed points, first A352520, rank stat A352513.
Cf. A088218, A115994, A238352, A350839, A352487, A352491.

pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pa[#]==k&]],{n,0,15},{k,0,n}]

T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k>i,x,1)*v[j-i])); r+=v); vector(#v, i, Vecrev(r[i], i))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023

Terms a(66) and beyond from Andrew Howroyd, Jan 19 2023

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 1, 11, 18, 3, 21, 35, 8, 41, 67, 20, 80, 131, 44, 1, 157, 257, 94, 4, 310, 505, 197, 12, 614, 996, 406, 32, 1218, 1973, 825, 80, 2421, 3915, 1669, 186, 1, 4819, 7781, 3364, 415, 5, 9602, 15486, 6762, 901, 17, 19147, 30855, 13567, 1918, 49

Offset: 0

Gus Wiseman, Mar 22 2022

			Triangle begins:
     1
     1
     1     1
     2     2
     3     5
     6     9     1
    11    18     3
    21    35     8
    41    67    20
    80   131    44     1
   157   257    94     4
   310   505   197    12
   614   996   406    32
For example, row n = 5 counts the following compositions:
  (113)    (5)     (23)
  (122)    (14)
  (1112)   (32)
  (1121)   (41)
  (1211)   (131)
  (11111)  (212)
           (221)
           (311)
           (2111)

Andrew Howroyd, Table of n, a(n) for n = 0..2507 (rows 0..200)
MathOverflow, Why 'excedances' of permutations? [closed].

The version for permutations is A008292, weak A123125.
Column k = 0 is A008930.
Row sums are A011782.
The opposite version for partitions is A114088.
The weak version for partitions is A115994.
Column k = 1 is A351983.
The corresponding rank statistic is A352516.
The opposite version is A352521, first col A219282, rank statistic A352514.
The weak opposite version is A352522, first col A238874, rank stat A352515.
The weak version is A352525, first col (k = 1) A177510, rank stat A352517.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352487 lists the excedance set of A122111, opposite A352490.
A352523 counts comps by unfixed points, first A352520, rank stat A352513.
Cf. A088218, A098825, A238352, A350839.

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

S(v,u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
T(n)={my(v=vector(1+n), s); v[1]=1; s=v; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1))); s+=v); [Vecrev(p) | p<-s]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 02 2023

A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2

Offset: 0

Gus Wiseman, Mar 22 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 83rd composition in standard order is (2,3,1,1), with strong nonexcedances {3,4}, so a(83) = 2.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A000225.
The weak version is A352515, counted by A352522 (first column A238874).
The opposite version is A352516, counted by A352524 (first column A008930).
The weak opposite version is A352517, counted by A352525 (first A177510).
The triangle A352521 counts these compositions (first column A219282).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed parts, first col A238351, rank stat A352512.
A352490 is the (strong) nonexcedance set of A122111.
A352523 counts comps by unfixed parts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[pa[stc[n]],{n,0,30}]

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 4, 4, 5, 4, 5, 5, 6, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 2, 3, 3

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A000225.
The strong version is A352514, counted by A352521 (first column A219282).
The strong opposite version is A352516, counted by A352524 (first A008930).
The opposite version is A352517, counted by A352525 (first column A177510).
Triangle A352522 counts these comps (first col A238874), partitions A115994.
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352488 is the weak nonexcedance set of A122111.
A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A131577, A238352.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
paw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[paw[stc[n]],{n,0,30}]

A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 5392th composition in standard order is (2,2,4,5), with excedances {1,3,4}, so a(5392) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A104462.
The opposite version is A352514, counted by A352521 (first column A219282).
The weak opposite version is A352515, counted by A352522 (first A238874).
The weak version is A352517, counted by A352525 (first column A177510).
The triangle A352524 counts these compositions (first column A008930).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352487 is the excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd[y_]:=Length[Select[Range[Length[y]],#

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positive positions of first appearances are A164894.
The version for partitions is A257990.
The strong opposite version is A352514, counted by A352521 (first A219282).
The opposite version is A352515, counted by A352522 (first column A238874).
The strong version is A352516, counted by A352524 (first column A008930).
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 is the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
Table[pdw[stc[n]],{n,0,30}]

A352520 Number of integer compositions y of n with exactly one nonfixed point y(i) != i.

Original entry on oeis.org

0, 0, 2, 1, 4, 5, 3, 7, 8, 9, 6, 11, 12, 13, 14, 10, 16, 17, 18, 19, 20, 15, 22, 23, 24, 25, 26, 27, 21, 29, 30, 31, 32, 33, 34, 35, 28, 37, 38, 39, 40, 41, 42, 43, 44, 36, 46, 47, 48, 49, 50, 51, 52, 53, 54, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 55, 67

Offset: 0

Gus Wiseman, Mar 29 2022

nonn

			The a(2) = 2 through a(8) = 8 compositions:
  (2)    (3)  (4)      (5)      (6)    (7)        (8)
  (1,1)       (1,3)    (1,4)    (1,5)  (1,6)      (1,7)
              (2,2)    (3,2)    (4,2)  (5,2)      (6,2)
              (1,2,1)  (1,1,3)         (1,2,4)    (1,2,5)
                       (1,2,2)         (1,3,3)    (1,4,3)
                                       (2,2,3)    (3,2,3)
                                       (1,2,3,1)  (1,2,1,4)
                                                  (1,2,3,2)

Compositions with no nonfixed points are counted by A010054.
The version for weak excedances is A177510.
Compositions with no fixed points are counted by A238351.
The version for fixed points is A240736.
This is column k = 1 of A352523.
A011782 counts compositions.
A238349 counts compositions by fixed points, rank stat A352512.
A352486 gives the nonfixed points of A122111, counted by A330644.
A352513 counts nonfixed points in standard compositions.
Cf. A008930, A088218, A098825, A114088, A115994, A219282, A238352, A238874, A350839.

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pnq[#]==1&]],{n,0,15}]

More terms from Alois P. Heinz, Mar 30 2022

A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that pi >= i.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 19, 25, 31, 38, 46, 59, 73, 92, 112, 135, 162, 196, 237, 289, 349, 417, 496, 587, 691, 820, 970, 1151, 1357, 1598, 1870, 2183, 2537, 2952, 3433, 3997, 4644, 5393, 6248, 7220, 8318, 9566, 10981, 12605, 14457, 16582, 19002, 21767, 24886, 28424, 32396, 36873, 41901, 47579, 53974, 61221

Offset: 0

Joerg Arndt, Mar 23 2014

nonn

			The a(13) = 31 such partitions of 13 are:
  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 2 3 6 ]
  14:  [ 2 2 4 5 ]
  15:  [ 2 2 9 ]
  16:  [ 2 3 3 5 ]
  17:  [ 2 3 4 4 ]
  18:  [ 2 3 8 ]
  19:  [ 2 4 7 ]
  20:  [ 2 5 6 ]
  21:  [ 2 11 ]
  22:  [ 3 3 3 4 ]
  23:  [ 3 3 7 ]
  24:  [ 3 4 6 ]
  25:  [ 3 5 5 ]
  26:  [ 3 10 ]
  27:  [ 4 4 5 ]
  28:  [ 4 9 ]
  29:  [ 5 8 ]
  30:  [ 6 7 ]
  31:  [ 13 ]

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 132 terms from Joerg Arndt)
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 5.

Cf. A219282 (superdiagonal compositions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions 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).

A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 47, 61, 80, 103, 138, 175, 224, 283, 362, 455, 577, 721, 898, 1111, 1380, 1701, 2106, 2577, 3156, 3844, 4680, 5671, 6879, 8312, 10034, 12060, 14478, 17319, 20715, 24703, 29442, 35004, 41578, 49247, 58278, 68796, 81132, 95502, 112320, 131877, 154705, 181158, 211908, 247475

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

Partitions with subdiagonal growth (A238876) with first part = 1.

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 9.

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).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions 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. A129176 and A227543.

\\ here b: nr parts; k: max part, b+w-1: partition sum.
seq(n)={my(M=matrix(n,1), v=vector(n+1)); M[1,1]=v[1]=v[2]=1; for(b=2, n, M=matrix(n-b+1,b,w,k, if(w>=k, sum(j=1, min(b-1,k), M[w+1-k,j]))); v+=concat(vector(b),vecsum(Vec(M))~)); v} \\ Andrew Howroyd, Jan 19 2024

N=55;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
x='x+O('x^N);
A(x) = sum(n=0, N, x^n * P(n) );
Vec(A(x)) \\ Joerg Arndt, Jan 23 2024

G.f.: Sum_{n>=0} x^n * P(n) where P(n) is the row polynomial of the n-th row of A129176. This works because A129176(j,k) is also the number of subdiagonal partitions of j+k with j parts. - John Tyler Rascoe, Jan 20 2024

A048285 Number of Dyck paths of length 2n with nondecreasing peaks.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 126, 316, 800, 2040, 5230, 13464, 34773, 90035, 233590, 607011, 1579438, 4114014, 10725109, 27979704, 73035818, 190737623, 498320800, 1302341411, 3404552915, 8902154847, 23281653957, 60897957049, 159312797657

Offset: 0

Olivier Roques (roques(AT)labri.u-bordeaux.fr)

The name refers to weakly increasing peaks. The case of strictly increasing peaks is counted by A008930. - David Callan, Feb 18 2004
a(n) ~ 0.11997*[(3+sqrt(5))/2]^n (Theorem 2 of the Penaud-Roques paper). - Emeric Deutsch, Mar 05 2008
Row sums of A138155. - Emeric Deutsch, Mar 05 2008
For a constant 0.1199765127480778967304984... see A239528. - Vaclav Kotesovec, Mar 21 2014

			a(3)=4 because we have UDUDUD, UDUUDD, UUDUDD and UUUDDD, where U=(1,1) and D=(1,-1).

Alois P. Heinz, Table of n, a(n) for n = 0..690 (terms n=1..300 from Vaclav Kotesovec)
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 21.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
J. G. Penaud and O. Roques, Génération de chemins de Dyck à pics croissants, Discrete Mathematics, Vol. 246, no. 1-3 (2002), 255-267.

Cf. A138155, A239528.

g:= 1+sum((-1)^n*z^(2*n+1)*(1-z)/(product((1-z)*(1-z^i)-z,i=1..n+1)), n=0..40): gser:=series(g,z=0,35): seq(coeff(gser,z,n),n=0..30); # Emeric Deutsch, Mar 05 2008
# second Maple program:
b:= proc(x, y, k, t) option remember; `if`(x=0, 1, `if`(y>0,
      `if`(t=1 and y>k, 0, b(x-1, y-1, `if`(t=1, min(k, y),
         k), 0)), 0) +`if`(y b(2*n, 0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 13 2017
# third Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, add(
      binomial(i, j)*add(b(n-2-(i-j)*2-2*t, i-j+t),
      t=0..n/2+j-i-1), j=0..i))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 13 2017

Table[SeriesCoefficient[Sum[(-1)^k*x^(2*k+1)*(1-x)/Product[(1-x)*(1-x^i)-x,{i,1,k+1}],{k,0,n}],{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Mar 21 2014 *)

G.f.: 1 + Sum_{n>=0} ((-1)^n x^{2n+1}(1-x)) / (Product_{i=1...n+1} ((1-x)(1-x^i)-x)).

Conjectural g.f.: Sum_{n>=0} (x*(1 - x))^n/( Product_{i=2..n+1} (1 - 2*x + x^i) ) (checked up to x^50). - Peter Bala, Mar 31 2017

More terms from Emeric Deutsch, Mar 05 2008

a(0)=1 prepended by Alois P. Heinz, Jan 31 2017

cp's OEIS Frontend

A352521 Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352520 Number of integer compositions y of n with exactly one nonfixed point y(i) != i.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that pi >= i.

Views

Author

Keywords

Examples

Links

Crossrefs

A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

Views