A048004 -id:A048004 - cp's OEIS frontend

A342527 Number of compositions of n with alternating parts equal.

1, 1, 2, 4, 6, 8, 11, 12, 16, 17, 21, 20, 29, 24, 31, 32, 38, 32, 46, 36, 51, 46, 51, 44, 69, 51, 61, 60, 73, 56, 87, 60, 84, 74, 81, 76, 110, 72, 91, 88, 115, 80, 123, 84, 117, 112, 111, 92, 153, 101, 132, 116, 139, 104, 159, 120, 161, 130, 141, 116, 205, 120, 151, 156, 178, 142, 195, 132, 183, 158

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (141)     (61)       (62)
                            (11111)  (222)     (151)      (71)
                                     (1212)    (232)      (161)
                                     (2121)    (313)      (242)
                                     (111111)  (12121)    (323)
                                               (1111111)  (1313)
                                                          (2222)
                                                          (3131)
                                                          (21212)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The odd-length case is A062968.
The even-length case is A065608.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts weakly decreasing is A342528.
A000005 counts constant compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A175342 counts compositions with constant differences.
A342495 counts compositions with constant first quotients.
A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(n) = 1 + n + A000203(n) - 2*A000005(n).

a(n) = A065608(n) + A062968(n).

A342495 Number of compositions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 10, 10, 11, 12, 12, 16, 16, 18, 20, 19, 18, 22, 22, 24, 28, 24, 24, 30, 27, 30, 30, 34, 30, 38, 36, 36, 36, 36, 40, 43, 40, 42, 46, 48, 42, 52, 46, 48, 52, 48, 48, 56, 55, 54, 54, 58, 54, 60, 58, 64, 64, 60, 60, 72, 64, 68, 74, 69, 72, 72

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
The a(1) = 1 through a(7) = 10 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (12)   (13)    (14)     (15)      (16)
             (21)   (22)    (23)     (24)      (25)
             (111)  (31)    (32)     (33)      (34)
                    (1111)  (41)     (42)      (43)
                            (11111)  (51)      (52)
                                     (222)     (61)
                                     (111111)  (124)
                                               (421)
                                               (1111111)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A175342.
The unordered version is A342496, ranked by A342522.
The strict unordered version is A342515.
The distinct version is A342529.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

a(n > 0) = 2*A342496(n) - A000005(n).

A342529 Number of compositions of n with distinct first quotients.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 19, 36, 67, 114, 197, 322, 564, 976, 1614, 2729, 4444, 7364, 12357, 20231, 33147, 53973, 87254, 140861, 227535, 368050, 589706, 940999, 1497912, 2378260, 3774297, 5964712, 9416411, 14822087, 23244440, 36420756

Offset: 0

Gus Wiseman, Mar 17 2021

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)    (3)    (4)      (5)
       (1,1)  (1,2)  (1,3)    (1,4)
              (2,1)  (2,2)    (2,3)
                     (3,1)    (3,2)
                     (1,1,2)  (4,1)
                     (1,2,1)  (1,1,3)
                     (2,1,1)  (1,2,2)
                              (1,3,1)
                              (2,1,2)
                              (2,2,1)
                              (3,1,1)
                              (1,1,2,1)
                              (1,2,1,1)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A325545.
The version for equal first quotients is A342495.
The unordered version is A342514, ranked by A342521.
The strict unordered version is A342520.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A003242, A008965, A048004, A059966, A167606, A253249, A318991, A318992, A342527, A342528.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

a(21)-a(35) from Alois P. Heinz, Jan 16 2025

A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 8, 13, 15, 16, 1, 13, 24, 29, 31, 32, 1, 21, 44, 56, 61, 63, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024

Offset: 1

N. J. A. Sloane, Mar 09 2007

Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k (see A048004). - N. J. A. Sloane, Apr 03 2011

Higher Order Fibonacci numbers: A126198(n,k) = Sum_{h=0..k} A048004(n,h); for example, A126198(7,3) = Sum_{h=0..3} A048004(7,h) or A126198(7,3) = 1 + 33 + 47 + 27 = 108, the 7th tetranacci number. A048004 row(7) produces A126198 row(7) list of 1,34,81,108,120,125,127,128 which are 1, the 7th Fibonacci, the 7th tribonacci, ... 7th octanacci numbers. - Richard Southern, Aug 04 2017

			Triangle begins:
  1;
  1,  2;
  1,  3,  4;
  1,  5,  7,  8;
  1,  8, 13, 15, 16;
  1, 13, 24, 29, 31, 32;
  1, 21, 44, 56, 61, 63, 64;
Could also be extended to a square array:
  1  1  1  1  1  1  1 ...
  1  2  2  2  2  2  2 ...
  1  3  4  4  4  4  4 ...
  1  5  7  8  8  8  8 ...
  1  8 13 15 16 16 16 ...
  1 13 24 29 31 32 32 ...
  1 21 44 56 61 63 64 ...
which when read by antidiagonals (downwards) gives A048887.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.

Alois P. Heinz, Rows n = 1..141

Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
Cf. A082267.

A126198 := proc(n,k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)),x=0,n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ",A126198(n,k)); od; od; # R. J. Mathar, Mar 09 2007
# second Maple program:
T:= proc(n, k) option remember;
      if n=0 or k=1 then 1
    else add(T(n-j, k), j=1..min(n, k))
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 23 2011

rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011, after Max Alekseyev *)

G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
T(n,3) = A008937(n) - A008937(n-3) for n>=3. T(n,4) = A107066(n-1) - A107066(n-5) for n>=5. T(n,5) = A001949(n+4) - A001949(n-1) for n>=5. - R. J. Mathar, Mar 09 2007
T(n,k) = A181695(n,k) - A181695(n-1,k). - Max Alekseyev, Nov 18 2010
Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - L. Edson Jeffery, Nov 29 2013

More terms from R. J. Mathar, Mar 09 2007

A119706 Total length of longest runs of 1's in all bitstrings of length n.

Original entry on oeis.org

1, 4, 11, 27, 62, 138, 300, 643, 1363, 2866, 5988, 12448, 25770, 53168, 109381, 224481, 459742, 939872, 1918418, 3910398, 7961064, 16190194, 32893738, 66772387, 135437649, 274518868, 556061298, 1125679616, 2277559414, 4605810806, 9309804278, 18809961926

Offset: 1

Adam Kertesz, Jun 09 2006, Jun 13 2006

nonn

a(n) divided by 2^n is the expected value of the longest run of heads in n tosses of a fair coin.

a(n) is also the sum of the number of binary words with at least one run of consecutive 0's of length >= i for i>=1. In other words A000225 + A008466 + A050231 + A050232 + ... . - Geoffrey Critzer, Jan 12 2013

			a(3)=11 because for the 8(2^3) possible runs 0 is longest run of heads once, 1 four times, 2 two times and 3 once and 0*1+1*4+2*2+3*1 = 11.

A. M. Odlyzko, Asymptotic Enumeration Methods, pp. 136-137
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 372.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.

Cf. A334833.

A038374 := proc(n) local nshft, thisr, resul; nshft := n ; resul :=0 ; thisr :=0 ; while nshft > 0 do if nshft mod 2 <> 0 then thisr := thisr+1 ; else resul := max(resul, thisr) ; thisr := 0 ; fi ; nshft := floor(nshft/2) ; od ; resul := max(resul, thisr) ; RETURN(resul) ; end : A119706 := proc(n) local count, c, rlen ; count := array(0..n) ; for c from 0 to n do count[c] := 0 ; od ; for c from 0 to 2^n-1 do rlen := A038374(c) ; count[rlen] := count[rlen]+1 ; od ; RETURN( sum('count[c]*c','c'=0..n) ); end: for n from 1 to 40 do print(n,A119706(n)) ; od : # R. J. Mathar, Jun 15 2006
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1,
      `if`(m=0, add(b(n-j, j), j=1..n),
      add(b(n-j, min(n-j, m)), j=1..min(n, m))))
    end:
a:= proc(n) option remember;
     `if`(n<2, n, 2*a(n-1) +b(n, 0))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 19 2014

nn=10;Drop[Apply[Plus,Table[CoefficientList[Series[1/(1-2x)-(1-x^n)/(1-2x+x^(n+1)),{x,0,nn}],x],{n,1,nn}]],1]  (* Geoffrey Critzer, Jan 12 2013 *)

a(n+1) = 2*a(n) + A007059(n+2)
a(n) > 2*a(n-1). a(n) = Sum_{i=1..(2^n)-1} A038374(i). - R. J. Mathar, Jun 15 2006
From Geoffrey Critzer, Jan 12 2013: (Start)
O.g.f.: Sum_{k>=1} 1/(1-2*x) - (1-x^k)/(1-2*x+x^(k+1)). - Corrected by Steven Finch, May 16 2020
a(n) = Sum_{k=1..n} A048004(n,k) * k.
(End)
Conjecture: a(n) = A102712(n+1)-2^n. - R. J. Mathar, Jun 05 2025

More terms from R. J. Mathar, Jun 15 2006

Name edited by Alois P. Heinz, Mar 18 2020

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

A342492 Number of compositions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 26, 37, 52, 73, 95, 125, 163, 208, 261, 330, 407, 498, 607, 734, 881, 1056, 1250, 1480, 1738, 2029, 2359, 2742, 3160, 3635, 4169, 4760, 5414, 6151, 6957, 7861, 8858, 9952, 11148, 12483, 13934, 15526, 17267, 19173, 21252, 23535, 25991

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

Also called log-concave-up compositions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (4,2,1,2,3) has first quotients (1/2,1/2,2,3/2) so is not counted under a(12), even though the first differences (-2,-1,1,1) are weakly increasing.
The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,2)    (4,1)        (4,2)
                       (2,1,1)    (1,1,3)      (5,1)
                       (1,1,1,1)  (2,1,2)      (1,1,4)
                                  (3,1,1)      (2,1,3)
                                  (1,1,1,2)    (2,2,2)
                                  (2,1,1,1)    (3,1,2)
                                  (1,1,1,1,1)  (4,1,1)
                                               (1,1,1,3)
                                               (2,1,1,2)
                                               (3,1,1,1)
                                               (1,1,1,1,2)
                                               (2,1,1,1,1)
                                               (1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325546.
The strictly increasing version is A342493.
The unordered version is A342497, ranked by A342523.
The strict unordered version is A342516.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A274199, A318991, A318992, A342527, A342528.

b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>=l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q >= l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(47) from Alois P. Heinz, Mar 25 2021

A342493 Number of compositions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 16, 22, 28, 39, 49, 61, 77, 93, 114, 140, 169, 198, 233, 276, 321, 381, 439, 509, 591, 678, 774, 883, 1007, 1147, 1300, 1465, 1641, 1845, 2068, 2317, 2590, 2881, 3193, 3549, 3928, 4341, 4793, 5282, 5813, 6401, 7027, 7699, 8432, 9221, 10076

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7).
The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)        (7)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)      (1,6)
              (2,1)  (2,2)    (2,3)    (2,4)      (2,5)
                     (3,1)    (3,2)    (3,3)      (3,4)
                     (1,1,2)  (4,1)    (4,2)      (4,3)
                     (2,1,1)  (1,1,3)  (5,1)      (5,2)
                              (2,1,2)  (1,1,4)    (6,1)
                              (3,1,1)  (2,1,3)    (1,1,5)
                                       (3,1,2)    (2,1,4)
                                       (4,1,1)    (2,2,3)
                                       (2,1,1,2)  (3,1,3)
                                                  (3,2,2)
                                                  (4,1,2)
                                                  (5,1,1)
                                                  (2,1,1,3)
                                                  (3,1,1,2)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A325547.
The weakly increasing version is A342492.
The strictly decreasing version is A342494.
The unordered version is A342498, ranked by A342524.
The strict unordered version is A342517.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A274199 counts compositions with all adjacent parts x < 2y.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q > l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 55] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(51) from Alois P. Heinz, Mar 18 2021

A105147 Triangular array read by rows: T(n,k) = number of compositions of n having smallest part equal to k.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 6, 1, 0, 1, 13, 2, 0, 0, 1, 27, 3, 1, 0, 0, 1, 56, 5, 2, 0, 0, 0, 1, 115, 9, 2, 1, 0, 0, 0, 1, 235, 15, 3, 2, 0, 0, 0, 0, 1, 478, 25, 5, 2, 1, 0, 0, 0, 0, 1, 969, 42, 8, 2, 2, 0, 0, 0, 0, 0, 1, 1959, 70, 12, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3952, 116, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Vladeta Jovovic, Apr 10 2005

			1;
1,  1;
3,  0, 1;
6,  1, 0, 1;
13, 2, 0, 0, 1;
27, 3, 1, 0, 0, 1;
56, 5, 2, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A048004.

Row sums give: A000079(n-1), columns k=1, 2 give: A099036(n-1), A200047. - Alois P. Heinz, Nov 13 2011

p:= (t, l)-> zip((x, y)->x+y, t, l, 0):
b:= proc(n) option remember; local j, t, h, m, s;
      t:= [0$(n-1), 1];
      for j to n-1 do
        h:= b(n-j);
        m:= nops(h);
        t:= p(p(t, [seq(h[i], i=1..min(j, m))]),
                   [0$(j-1), add(h[i], i=j+1..m)])
      od; t
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..15); # Alois P. Heinz, Nov 13 2011

zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; p[t_, l_] := zip[Plus, t, l, 0]; b[n_] := b[n] = Module[{j, t, h, m, s}, t = Append[Array[0&, n-1], 1]; For[j = 1, j <= n-1 , j++, h = b[n-j]; m = Length[h]; t = p[p[t, h[[1 ;; Min[j, m]]]], Append[Array[0&, j-1], h[[Min[j, m]+1 ;; m]] // Total]]]; t]; Table[b[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

G.f. for k-th column: (1-x)^2*x^k/((1-x-x^k)*(1-x-x^(k+1))).

A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 8, 7, 4, 2, 1, 1, 13, 13, 8, 4, 2, 1, 1, 21, 24, 15, 8, 4, 2, 1, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 144, 274, 208, 120, 63, 32, 16, 8, 4, 2, 1, 1, 233, 504, 401, 236, 125, 64, 32, 16, 8, 4, 2, 1

Offset: 2

Roger L. Bagula, Dec 03 2010

Antidiagonal sums are A048888. This is a transposed version of A048887, so the bivariate generating function is obtained by swapping the two arguments.

Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

			The array starts in row n=2 with columns k >= 1 as:
  1   1   1   1   1   1   1   1   1   1
  1   2   2   2   2   2   2   2   2   2
  1   3   4   4   4   4   4   4   4   4
  1   5   7   8   8   8   8   8   8   8
  1   8  13  15  16  16  16  16  16  16
  1  13  24  29  31  32  32  32  32  32
  1  21  44  56  61  63  64  64  64  64
  1  34  81 108 120 125 127 128 128 128
  1  55 149 208 236 248 253 255 256 256

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 125, 155.

Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014

Cf. A000045, A048887, A048004, A126198.

A092921 := proc(n,k) if k <= 0 or n <= 0 then 0; elif k = 1 or n = 1 then 1; else add( procname(n-i,k),i=1..k) ; end if; end proc:
A175331 := proc(n,k) A092921(n,k) ; end proc: # R. J. Mathar, Dec 17 2010

f[x_, n_] = (x - x^(m + 1))/(1 - 2*x + x^(m + 1))
a = Table[Table[SeriesCoefficient[
      Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
Flatten[%]

T(n,k) = A092921(n,k), n >= 2.
T(n,2) = A000045(n).
T(n,3) = A000073(n+2).
T(n,4) = A000078(n+2).

cp's OEIS Frontend

A342527 Number of compositions of n with alternating parts equal.

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A342495 Number of compositions of n with constant (equal) first quotients.

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A342529 Number of compositions of n with distinct first quotients.

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A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

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A119706 Total length of longest runs of 1's in all bitstrings of length n.

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A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

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A342492 Number of compositions of n with weakly increasing first quotients.

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