A216652 -id:A216652 - cp's OEIS frontend

A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

1, 1, 0, 2, 2, 4, 8, 6, 16, 12, 22, 40, 40, 66, 48, 74, 74, 154, 210, 228, 242, 240, 286, 394, 806, 536, 840, 654, 1146, 1618, 2036, 2550, 2212, 2006, 2662, 4578, 4170, 7122, 4842, 6012, 6214, 11638, 13560, 16488, 14738, 15444, 16528, 25006, 41002, 32802

Offset: 0

Gus Wiseman, Sep 18 2020

nonn

			The a(1) = 1 through a(9) = 12 compositions (empty column shown as dot):
   (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)    (1,8)
           (2,1)  (3,1)  (2,3)  (5,1)    (2,5)  (3,5)    (2,7)
                         (3,2)  (1,2,3)  (3,4)  (5,3)    (4,5)
                         (4,1)  (1,3,2)  (4,3)  (7,1)    (5,4)
                                (2,1,3)  (5,2)  (1,2,5)  (7,2)
                                (2,3,1)  (6,1)  (1,3,4)  (8,1)
                                (3,1,2)         (1,4,3)  (1,3,5)
                                (3,2,1)         (1,5,2)  (1,5,3)
                                                (2,1,5)  (3,1,5)
                                                (2,5,1)  (3,5,1)
                                                (3,1,4)  (5,1,3)
                                                (3,4,1)  (5,3,1)
                                                (4,1,3)
                                                (4,3,1)
                                                (5,1,2)
                                                (5,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

A072706 counts unimodal strict compositions.
A220377*6 counts these compositions of length 3.
A305713 is the unordered version.
A337462 is the not necessarily strict version.
A000740 counts relatively prime compositions, with strict case A332004.
A051424 counts pairwise coprime or singleton partitions.
A101268 considers all singletons to be coprime, with strict case A337562.
A178472 counts compositions with a common factor > 1.
A327516 counts pairwise coprime partitions, with strict case A305713.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A007359, A007360, A087087, A216652, A220377, A302569, A307719, A326675, A333227, A337461.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||UnsameQ@@#&&CoprimeQ@@#&]],{n,0,10}]

a(n) = A337562(n) - 1 for n > 1.

A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 1, 6, 12, 0, 0, 0, 0, 0, 1, 8, 18, 0, 0, 0, 0, 0, 0, 1, 8, 24, 24, 0, 0, 0, 0, 0, 0, 1, 10, 30, 24, 0, 0, 0, 0, 0, 0, 0, 1, 10, 42, 48, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 48, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 60, 120, 0

Offset: 1

Henry Bottomley, Jun 21 2002

If terms in the compositions did not need to be distinct then the triangle would have values C(n-1,k-1), essentially A007318 offset.

			T(6,2)=4 since 6 can be written as 1+5=2+4=4+2=5+1.
Triangle starts (trailing zeros omitted for n>=10):
[ 1]  1;
[ 2]  1, 0;
[ 3]  1, 2, 0;
[ 4]  1, 2, 0, 0;
[ 5]  1, 4, 0, 0, 0;
[ 6]  1, 4, 6, 0, 0, 0;
[ 7]  1, 6, 6, 0, 0, 0, 0;
[ 8]  1, 6, 12, 0, 0, 0, 0, 0;
[ 9]  1, 8, 18, 0, 0, 0, 0, 0, 0;
[10]  1, 8, 24, 24, 0, 0, ...;
[11]  1, 10, 30, 24, 0, 0, ...;
[12]  1, 10, 42, 48, 0, 0, ...;
[13]  1, 12, 48, 72, 0, 0, ...;
[14]  1, 12, 60, 120, 0, 0, ...;
[15]  1, 14, 72, 144, 120, 0, 0, ...;
[16]  1, 14, 84, 216, 120, 0, 0, ...;
[17]  1, 16, 96, 264, 240, 0, 0, ...;
[18]  1, 16, 114, 360, 360, 0, 0, ...;
[19]  1, 18, 126, 432, 600, 0, 0, ...;
[20]  1, 18, 144, 552, 840, 0, 0, ...;
These rows (without the zeros) are shown in the Richmond/Knopfmacher reference.
From _Gus Wiseman_, Oct 17 2022: (Start)
Column n = 8 counts the following compositions.
  (8)  (1,7)  (1,2,5)
       (2,6)  (1,3,4)
       (3,5)  (1,4,3)
       (5,3)  (1,5,2)
       (6,2)  (2,1,5)
       (7,1)  (2,5,1)
              (3,1,4)
              (3,4,1)
              (4,1,3)
              (4,3,1)
              (5,1,2)
              (5,2,1)
(End)

Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..100, flattened).
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.
Index entries for sequences related to compositions

Columns (offset) include A057427 and A052928.
Row sums are A032020.
Cf. A060016, A072576.
A008289 is the version for partitions (zeros removed).
A072575 counts strict compositions by maximum.
A097805 is the non-strict version, or A007318 (zeros removed).
A113704 is the constant instead of strict version.
A216652 is a condensed version (zeros removed).
A336131 counts splittings of partitions with distinct sums.
A336139 counts strict compositions of each part of a strict composition.
Cf. A075900, A097910, A307068, A336127, A336128, A336130, A336132, A336141, A336142, A336342, A336343.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],Length[#]==k&]],{n,0,15},{k,1,n}] (* Gus Wiseman, Oct 17 2022 *)

N=21;  q='q+O('q^N);
gf=sum(n=0,N, n! * z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
    p = Pol(polcoeff(P, n),'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n,k,v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

T(n, k) = T(n-k, k)+k*T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise] = A000142(k)*A060016(n, k).

G.f.: sum(n>=0, n! * z^n * q^((n^2+n)/2) / prod(k=1..n, 1-q^k ) ), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A032020. [Joerg Arndt, Oct 20 2012]

A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 12, 16, 24, 52, 64, 88, 132, 180, 344, 416, 616, 816, 1176, 1496, 2736, 3232, 4756, 6176, 8756, 11172, 15576, 24120, 30460, 41456, 55740, 74440, 97976, 130192, 168408, 256464, 315972, 429888, 558192, 749920, 958264, 1274928, 1621272, 2120288, 3020256

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

Moebius transform of A032020.

Ranking these compositions using standard compositions (A066099) gives the intersection of A233564 (strict) with A291166 (relatively prime). - Gus Wiseman, Oct 18 2020

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].
From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(8) = 16 compositions (empty column indicated by dot):
  (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
          (2,1)  (3,1)  (2,3)  (5,1)    (2,5)    (3,5)
                        (3,2)  (1,2,3)  (3,4)    (5,3)
                        (4,1)  (1,3,2)  (4,3)    (7,1)
                               (2,1,3)  (5,2)    (1,2,5)
                               (2,3,1)  (6,1)    (1,3,4)
                               (3,1,2)  (1,2,4)  (1,4,3)
                               (3,2,1)  (1,4,2)  (1,5,2)
                                        (2,1,4)  (2,1,5)
                                        (2,4,1)  (2,5,1)
                                        (4,1,2)  (3,1,4)
                                        (4,2,1)  (3,4,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A007360, A032020, A108700, A302698.
A000740 is the non-strict version.
A078374 is the unordered version (non-strict: A000837).
A101271*6 counts these compositions of length 3 (non-strict: A000741).
A337561/A337562 is the pairwise coprime instead of relatively prime version (non-strict: A337462/A101268).
A289509 gives the Heinz numbers of relatively prime partitions.
A333227/A335235 ranks pairwise coprime compositions.
Cf. A001523, A178472, A216652, A289508, A291166, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&GCD@@#<=1&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 2, 7, 18, 45, 101, 219, 461, 957, 1957, 3978, 8036, 16182, 32506, 65202, 130642, 261601, 523598, 1047709, 2096062, 4192946, 8386912, 16775117, 33551832, 67105663, 134213789, 268430636, 536865013, 1073734643, 2147474910, 4294956706, 8589921771

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(4) = 2 through a(4) = 18 compositions:
  (112)  (113)   (114)
  (121)  (122)   (132)
         (131)   (141)
         (212)   (213)
         (1112)  (231)
         (1121)  (312)
         (1211)  (1113)
                 (1122)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11112)
                 (11121)
                 (11211)
                 (12111)

Ranked by the complement of the intersection of A114994 and A333255.
A128422 counts only the case of length 3.
A218004 counts the complement.
A332834 is the weak version.
A337481 is the strict version.
A001523 counts unimodal compositions, with complement counted by A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A332745/A332835 count partitions/compositions with weakly increasing or weakly decreasing run-lengths.
Cf. A216652, A329398, A332831, A332833, A337462, A337483, A337484, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!GreaterEqual@@#&]],{n,0,15}]

a(n) = 2^(n-1) - A000009(n) - A000041(n) + 1, n > 0.

A291960 Triangle read by rows: T(n,k) = T(n-k,k-1) + k * T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 3, 1, 0, 1, 7, 1, 0, 1, 7, 3, 0, 1, 15, 6, 0, 1, 15, 10, 1, 0, 1, 31, 16, 1, 0, 1, 31, 33, 3, 0, 1, 63, 45, 6, 0, 1, 63, 79, 14, 0, 1, 127, 130, 20, 1, 0, 1, 127, 198, 45, 1, 0, 1, 255, 300, 69, 3, 0, 1, 255, 517, 135

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1,  1;
  0, 1,  1;
  0, 1,  3;
  0, 1,  3,  1;
  0, 1,  7,  1;
  0, 1,  7,  3;
  0, 1, 15,  6;
  0, 1, 15, 10, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A204856.
Columns 0-2 give A000007, A000012, A052551(n-3).
Cf. A008289, A216652, A291969.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-j*x^j).

A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(2) = 1 through a(5) = 11 compositions:
  (11)  (111)  (22)    (113)
               (112)   (122)
               (121)   (131)
               (211)   (212)
               (1111)  (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)

Ranked by the complement of the intersection of A333255 and A333256.
A332834 is the weak version.
A337482 is the semi-strict version.
A337484 counts only compositions of length 3.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A218004 counts strictly increasing or weakly decreasing compositions.
Cf. A216652, A329398, A337462, A337483, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2^(n-1) - 2*A000009(n) + 1, n > 0.

A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 6, 3, 16, 5, 6, 1, 1, 4, 15, 4, 25, 6, 7, 1, 1, 6, 9, 28, 5, 36, 7, 8, 1, 1, 4, 21, 16, 45, 6, 49, 8, 9, 1, 1, 10, 9, 52, 25, 66, 7, 64, 9, 10, 1, 1, 4, 39, 16, 105, 36, 91, 8, 81, 10, 11, 1, 1, 12, 9, 100, 25, 186, 49, 120, 9, 100, 11, 12, 1, 1, 6, 45, 16, 205, 36, 301, 64, 153, 10, 121, 12, 13, 1

Offset: 1

N. J. A. Sloane, Mar 05 2017

See A101391 for the triangle T(n,k) = number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2 <= k <= n).

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  3,   1;
  1,  4,  3,   4,   1;
  1,  2,  9,   4,   5,   1;
  1,  6,  3,  16,   5,   6,  1;
  1,  4, 15,   4,  25,   6,  7,   1;
  1,  6,  9,  28,   5,  36,  7,   8,  1;
  1,  4, 21,  16,  45,   6, 49,   8,  9,   1;
  1, 10,  9,  52,  25,  66,  7,  64,  9,  10,  1;
  1,  4, 39,  16, 105,  36, 91,   8, 81,  10, 11,  1;
  1, 12,  9, 100,  25, 186, 49, 120,  9, 100, 11, 12, 1;
  ...
From _Gus Wiseman_, Nov 12 2020: (Start)
Row n = 6 counts the following compositions:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (51)  (123)  (1131)  (11121)
             (132)  (1311)  (11211)
             (141)  (3111)  (12111)
             (213)          (21111)
             (231)
             (312)
             (321)
             (411)
(End)

Alois P. Heinz, Rows n = 1..200, flattened (first 100 rows from Chai Wah Wu)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

A072704 counts the unimodal instead of coprime version.
A087087 and A335235 rank these compositions.
A101268 gives row sums.
A101391 is the relatively prime instead of pairwise coprime version.
A282749 is the unordered version.
A000740 counts relatively prime compositions, with strict case A332004.
A007360 counts pairwise coprime or singleton strict partitions.
A051424 counts pairwise coprime or singleton partitions, ranked by A302569.
A097805 counts compositions by sum and length.
A178472 counts compositions with a common divisor.
A216652 and A072574 count strict compositions by sum and length.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A335235 ranks pairwise coprime or singleton compositions.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337562 counts pairwise coprime or singleton strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
Cf. A000837, A007359, A302568, A337461, A337561, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* Gus Wiseman, Nov 12 2020 *)

It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.

A291968 Triangle read by rows: T(n,k) = (k+1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 2, 6, 0, 2, 6, 0, 2, 12, 0, 2, 12, 24, 0, 2, 18, 24, 0, 2, 18, 48, 0, 2, 24, 72, 0, 2, 24, 96, 120, 0, 2, 30, 120, 120, 0, 2, 30, 168, 240, 0, 2, 36, 192, 360, 0, 2, 36, 240, 600, 0, 2, 42, 288, 720, 720, 0, 2, 42, 336, 1080, 720, 0, 2, 48, 384

Offset: 0

Seiichi Manyama, Sep 07 2017

			First few rows are:
  1;
  0, 2;
  0, 2;
  0, 2,  6;
  0, 2,  6;
  0, 2, 12;
  0, 2, 12, 24;
  0, 2, 18, 24;
  0, 2, 18, 48;
  0, 2, 24, 72;
  0, 2, 24, 96, 120.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A072576.
Columns 0-1 give A000007, A007395.
Cf. A216652.

G.f. of column k: (k+1)! * x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

A362208 Irregular triangle read by rows: T(n, k) is the number of compositions (ordered partitions) of n into exactly k distinct parts between the members of [k^2].

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 4, 0, 2, 6, 0, 2, 6, 0, 0, 12, 0, 0, 18, 0, 0, 24, 24, 0, 0, 30, 24, 0, 0, 42, 48, 0, 0, 42, 72, 0, 0, 48, 120, 0, 0, 48, 144, 120, 0, 0, 48, 216, 120, 0, 0, 42, 264, 240, 0, 0, 42, 360, 360, 0, 0, 30, 432, 600, 0, 0, 24, 552, 840, 0, 0, 18, 648, 1200, 720

Offset: 1

Stefano Spezia, Apr 11 2023

			The irregular triangle begins:
    1;
    0;
    0, 2;
    0, 2;
    0, 4;
    0, 2,  6;
    0, 2,  6;
    0, 0, 12;
    0, 0, 18;
    0, 0, 24,  24;
    0, 0, 30,  24;
    0, 0, 42,  48;
    0, 0, 42,  72;
    0, 0, 48, 120;
    0, 0, 48, 144, 120;
    ...
T(7,3) = 6 since we have: 1+2+4, 1+4+2, 2+1+4, 2+4+1, 4+1+2, 4+2+1.

Cf. A000290, A003056 (row lengths), A072574, A216652.

Cf. A362209, A362221 (unordered partitions).

Flatten[Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n,All,Range[k^2]], UnsameQ@@#&], Length[#]==k&]], {n, 21}, {k, Floor[(Sqrt[8n+1]-1)/2]}]] (* After Gus Wiseman in A072574 *)

cp's OEIS Frontend

A337561 Number of pairwise coprime strict compositions of n, where a singleton is not considered coprime unless it is (1).

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A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n.

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A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

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A337451 Number of relatively prime strict compositions of n with no 1's.

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A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

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A291960 Triangle read by rows: T(n,k) = T(n-k,k-1) + k * T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

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A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

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