A219282 -id:A219282 - cp's OEIS frontend

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

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).

A063978 Sum_{i for which n - i(i-1)/2 >= 0} binomial (n - i(i-1)/2, i).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 27, 40, 58, 83, 118, 167, 235, 328, 454, 624, 853, 1161, 1574, 2125, 2856, 3821, 5090, 6754, 8931, 11773, 15474, 20280, 26502, 34533, 44870, 58142, 75145, 96885, 124630, 159973, 204909, 261930, 334143, 425417, 540566, 685576, 867885, 1096726, 1383545, 1742509, 2191123, 2750980

Offset: 0

Helmut Schnitzspan (HSchnitzspan(AT)gmx.de), Sep 05 2001

nonn

Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k)>=k-1 for k>=2, see example. - Joerg Arndt, Dec 19 2012

			From _Joerg Arndt_, Dec 19 2012: (Start)
The a(7) = 18 compositions 7 = p(1) + p(2) + ... + p(m) such that for k>=2 p(k)>=k-1 are
[ 1]  [ 1 1 2 3 ]
[ 2]  [ 1 1 5 ]
[ 3]  [ 1 2 4 ]
[ 4]  [ 1 3 3 ]
[ 5]  [ 1 4 2 ]
[ 6]  [ 1 6 ]
[ 7]  [ 2 1 4 ]
[ 8]  [ 2 2 3 ]
[ 9]  [ 2 3 2 ]
[10]  [ 2 5 ]
[11]  [ 3 1 3 ]
[12]  [ 3 2 2 ]
[13]  [ 3 4 ]
[14]  [ 4 1 2 ]
[15]  [ 4 3 ]
[16]  [ 5 2 ]
[17]  [ 6 1 ]
[18]  [ 7 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Cf. A064188.

Cf. A219282 (compositions such that p(k)>=k for all k; superdiagonal bargraphs).

b:= proc(n, i) option remember; `if`(n=0, 1,
      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 28 2014

Table[ Sum[ Binomial[ n - i(i - 1)/2, i], {i, 0, Floor[ (Sqrt[8n + 1] - 1)/2]} ], {n, 0, 40}]

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

G.f.: Sum_{k>=1} x^(k*(k-1)/2) / (1-x)^k. - Vladeta Jovovic, Sep 25 2004

More terms from Dean Hickerson, Sep 06 2001

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

A352875 Number of integer compositions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 1, 2, 5, 10, 21, 42, 86, 174, 351, 708, 1424, 2861, 5743, 11520, 23092, 46269, 92673, 185562, 371469, 743491, 1487870, 2977164, 5956616, 11916910, 23839736, 47688994, 95393322, 190811346, 381662507, 763389209, 1526881959, 3053930971, 6108131542, 12216698288

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
  .  (1)  (11)  (12)   (13)    (14)
                (111)  (22)    (32)
                       (112)   (113)
                       (121)   (122)
                       (1111)  (131)
                               (221)
                               (1112)
                               (1121)
                               (1211)
                               (11111)

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

The version for partitions is A001522, ranked by A352827 (unproved).
The version for permutations is A002467, complement A000166.
The complement for partitions is A064428, ranked by A352826 (unproved).
This is the sum of latter columns of A238349, nonfixed A352523.
The complement is counted by A238351.
The complement for reversed partitions is A238394, ranked by A352830.
The version for reversed partitions is A238395, ranked by A352872.
The case of just one fixed point is A240736.
A008290 counts permutations by fixed points, nonfixed A098825.
A011782 counts compositions.
A115720 and A115994 count partitions by Durfee square.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352512 counts fixed points in standard compositions, nonfixed A352513.
A352521 = comps by subdiags, first col A219282, rank stat A352514.
A352522 = comps by weak subdiags, first col A238874, rank stat A352515.
A352524 = comps by superdiags, first col A008930, rank stat A352516.
A352525 = comps by weak superdiags, col k=1 A177510, rank stat A352517.
A352833 counts partitions by fixed points.
Cf. A006918, A008292, A088218, A114088, A123125, A188674, A257990.

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

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

a(n) = 2^(n-1) - A238351(n) for n >= 1. - Andrew Howroyd, Jan 02 2023

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

A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 1, 1, 12, 2, 1, 16, 5, 1, 20, 9, 1, 25, 16, 1, 30, 25, 1, 36, 39, 1, 1, 42, 56, 2, 1, 49, 80, 5, 1, 56, 109, 10, 1, 64, 147, 19, 1, 72, 192, 32, 1, 81, 249, 54, 1, 90, 315, 84, 1, 100, 396, 129, 1, 1, 110, 489, 190, 2, 1, 121, 600, 275, 5

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1   1
   1   2
   1   4
   1   6
   1   9   1
   1  12   2
   1  16   5
   1  20   9
   1  25  16
   1  30  25
   1  36  39   1
   1  42  56   2
   1  49  80   5
   1  56 109  10
For example, row n = 7 counts the following partitions:
  (1111111)  (7)       (43)
             (52)      (331)
             (61)
             (322)
             (421)
             (511)
             (2221)
             (3211)
             (4111)
             (22111)
             (31111)
             (211111)

Row sums are A000041.
Row lengths are A000194, reversed A003056.
Column k = 1 is A002620, reversed A238875.
Column k = 2 is A097701.
The version for permutations is A008292, opposite A123125.
The weak version is A115720/A115994, rank statistic A257990.
The version for compositions is A352524, weak A352525.
The version for reversed partitions is A353319.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

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

A305556 Irregular triangle read by rows: T(n,k) is the number of superdiagonal bargraphs with area n and with k columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 5, 3, 1, 6, 6, 1, 7, 10, 1, 8, 15, 1, 1, 9, 21, 4, 1, 10, 28, 10, 1, 11, 36, 20, 1, 12, 45, 35, 1, 13, 55, 56, 1, 1, 14, 66, 84, 5, 1, 15, 78, 120, 15, 1, 16, 91, 165, 35, 1, 17, 105, 220, 70, 1, 18, 120, 286, 126, 1, 19, 136, 364, 210, 1, 1, 20, 153, 455

Offset: 1

R. J. Mathar, Jun 21 2018

			   1,
   1,
   1,   1,
   1,   2,
   1,   3,
   1,   4,   1,
   1,   5,   3,
   1,   6,   6,
   1,   7,  10,
   1,   8,  15,   1,
   1,   9,  21,   4,
   1,  10,  28,  10,
   1,  11,  36,  20,
   1,  12,  45,  35,
   1,  13,  55,  56,   1,
   1,  14,  66,  84,   5,
   1,  15,  78, 120,  15,
   1,  16,  91, 165,  35,
   1,  17, 105, 220,  70,
   1,  18, 120, 286, 126,
   1,  19, 136, 364, 210,   1,
   1,  20, 153, 455, 330,   6,
   1,  21, 171, 560, 495,  21,
   1,  22, 190, 680, 715,  56,
   1,  23, 210, 816,1001, 126,
   1,  24, 231, 969,1365, 252,
   1,  25, 253,1140,1820, 462,
   1,  26, 276,1330,2380, 792,   1,
   1,  27, 300,1540,3060,1287,   7,
   1,  28, 325,1771,3876,2002,  28,

Emeric Deutsch, Emanuele Munarini, Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Cf. A219282 (row sums), A000217 (column 3), A000292 (column 4), A000332 (column 5)

A305556 := proc(n,k)
    binomial(n-binomial(k,2)-1,k-1) ;
end proc:
for n from 0 to 30 do
for k from 1 to floor((sqrt(1+8*n)-1)/2) do
    printf("%d,",A305556(n,k)) ;
end do:
end do:

T(n,k) = binomial(n-1-k*(k-1)/2,k-1), 1<=k <= (sqrt(1+8*n)-1)/2.

A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 2, 1, 5, 4, 2, 7, 6, 2, 10, 6, 6, 15, 7, 7, 1, 18, 14, 7, 3, 26, 15, 11, 4, 35, 17, 19, 6, 47, 24, 19, 11, 61, 33, 22, 18, 1, 80, 44, 28, 20, 4, 103, 54, 42, 25, 7, 138, 60, 57, 31, 11, 175, 85, 58, 52, 15, 224, 112, 66, 64, 24

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1  1
   2  1
   2  3
   4  2  1
   5  4  2
   7  6  2
  10  6  6
  15  7  7  1
  18 14  7  3
  26 15 11  4
  35 17 19  6
  47 24 19 11
  61 33 22 18  1
  80 44 28 20  4
For example, row n = 9 counts the following reversed partitions:
  (1134)       (9)     (27)   (234)
  (1224)       (18)    (36)
  (1233)       (117)   (45)
  (11115)      (126)   (135)
  (11124)      (1116)  (144)
  (11133)      (1125)  (225)
  (11223)      (2223)  (333)
  (12222)
  (111114)
  (111123)
  (111222)
  (1111113)
  (1111122)
  (11111112)
  (111111111)

Row sums are A000041.
Row lengths are A003056.
The version for permutations is A008292, opposite A123125.
The weak unreversed version is A115720/A115994, rank statistic A257990.
For fixed points instead of excedances we have A238352, rank stat A352822.
Column k = 0 is A238875.
The version for compositions is A352524, weak A352525.
The version for unreversed partitions is A353318.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
Cf. A000701, A002620, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

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

A236310 Expansion of Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 52, 62, 74, 89, 108, 132, 162, 199, 244, 298, 362, 437, 524, 625, 743, 882, 1047, 1244, 1480, 1763, 2102, 2507, 2989, 3560, 4233, 5022, 5943, 7015, 8261, 9709, 11393, 13354, 15641, 18312, 21435, 25089, 29365, 34367, 40213, 47036, 54985, 64227, 74950

Offset: 0

Joerg Arndt, Apr 22 2014

nonn

a(n) is the number of compositions of n such that the first part is equal to the number of parts and all parts are greater than or equal to the first part. - John Tyler Rascoe, Feb 10 2024

			From _John Tyler Rascoe_, Feb 10 2024: (Start)
The compositions for n = 9..11 are:
9:  [3,3,3], [2,7];
10: [3,4,3], [3,3,4], [2,8];
11: [3,4,4], [3,3,5], [3,5,3], [2,9].
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A098131 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).
Cf. A219282 (g.f. Sum_{k>=0} x^(k*(k+1)/2)/(1-x)^k).
Cf. A063978 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^(k+1)).

N=66; q='q+O('q^N);
gf=sum(n=0, N, q^((n+1)^2) / (1-q)^n );
concat([0],Vec(gf))

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

G.f.: Sum_{k>0} A(x,k) where A(x,k) = (x^k)*(x^k/(1-x))^(k-1) is the g.f. for compositions of this kind with first part k. - John Tyler Rascoe, Feb 10 2024

cp's OEIS Frontend

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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A063978 Sum_{i for which n - i*(i-1)/2 >= 0} binomial (n - i*(i-1)/2, i).

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

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A352875 Number of integer compositions y of n with a fixed point y(i) = i.

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A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

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A305556 Irregular triangle read by rows: T(n,k) is the number of superdiagonal bargraphs with area n and with k columns.

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A063978 Sum_{i for which n - i(i-1)/2 >= 0} binomial (n - i(i-1)/2, i).