A240026 -id:A240026 - cp's OEIS frontend

A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 5, 1, 0, 6, 0, 2, 7, 2, 0, 8, 2, 2, 9, 3, 0, 13, 0, 2, 11, 3, 4, 15, 0, 3, 13, 6, 0, 18, 0, 4, 20, 4, 0, 19, 2, 8, 18, 5, 0, 23, 6, 6, 20, 5, 0, 30, 0, 5, 25, 6, 7, 29, 0, 6, 24, 15, 0, 32, 0, 6, 34, 7, 4, 34, 0, 14, 31, 7, 0, 39, 9, 7, 31, 9, 0, 49, 5, 9, 33, 8, 10, 42, 0, 12

Offset: 1

Clark Kimberling

nonn

			E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.

Antti Karttunen, Table of n, a(n) for n = 1..12580 (first 1000 terms from Fausto A. C. Cariboni)
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000217, A007862, A014406, A014407, A023645, A047966, A049982, A049983, A049986, A049987, A049992, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

a(n)= t=0; st=0; forstep(s=(n-3)\3,1,-1, st++; for(c=1,st, m=3; w=m*(s+c); while(wRick L. Shepherd, Aug 30 2006

G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

a(n) = A049992(n) - A023645(n). - Antti Karttunen, Feb 20 2023

A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A007294.
This sequence and A025487, considered as sets, are related by the partition conjugation function A122111(.), which maps the members of either set 1:1 onto the other set. - Peter Munn, Feb 10 2022

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   30: {1,2,3}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.

Related to A025487 via A122111.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],OrderedQ[Differences[Append[primeptn[#],0]]]&]

A325360 Heinz numbers of integer partitions whose differences are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A240026.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Robert Price, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A056239, A112798, A240026, A325328, A325352, A325354, A325360, A325361, A325368, A325394, A325400.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],OrderedQ[Differences[primeptn[#]]]&]

A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 7, 8, 13, 14, 14, 20, 20, 22, 29, 31, 31, 39, 41, 43, 52, 55, 55, 68, 68, 70, 81, 84, 88, 103, 103, 106, 119, 125, 125, 143, 143, 147, 167, 171, 171, 190, 192, 200, 218, 223, 223, 246, 252, 258, 278, 283, 283, 313, 313, 318, 343, 349, 356, 385, 385

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(8) = 1 because we have only the following strictly increasing arithmetic progression of positive integers with at least 3 terms and sum <= 8: 1+2+3.
a(9) = 3 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 9: 1+2+3, 1+3+5, and 2+3+4.
a(10) = 4 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 10: 1+2+3, 1+3+5, 2+3+4, and 1+2+3+4.
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007862, A014405, A047966, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

a(n) = Sum_{k=1..n} A014405(k). - Sean A. Irvine, Oct 22 2018

G.f.: (g.f. of A014405)/(1-x). - Petros Hadjicostas, Sep 29 2019

a(59)-a(67) corrected by Fausto A. C. Cariboni, Oct 02 2018

A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 14, 15, 19, 21, 22, 28, 30, 32, 39, 42, 44, 54, 58, 61, 72, 77, 82, 96, 102, 108, 124, 133, 141, 160, 171, 180, 203, 218, 230, 256, 273, 289, 320, 342, 361, 395, 423, 447, 486, 520, 548, 594, 635, 669, 721, 769, 811, 871, 928, 978, 1044, 1114

Offset: 0

Joerg Arndt, Jan 05 2011

nonn

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) < p(k) - p(k-1) for all k >= 3.

			There are a(17) = 21 such partitions of 17:
01:  [ 1 2 4 10 ]
02:  [ 1 2 5 9 ]
03:  [ 1 2 14 ]
04:  [ 1 3 13 ]
05:  [ 1 4 12 ]
06:  [ 1 5 11 ]
07:  [ 1 16 ]
08:  [ 2 3 12 ]
09:  [ 2 4 11 ]
10:  [ 2 5 10 ]
11:  [ 2 15 ]
12:  [ 3 4 10 ]
13:  [ 3 5 9 ]
14:  [ 3 14 ]
15:  [ 4 5 8 ]
16:  [ 4 13 ]
17:  [ 5 12 ]
18:  [ 6 11 ]
19:  [ 7 10 ]
20:  [ 8 9 ]
21:  [ 17 ]
- _Joerg Arndt_, Mar 31 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 225 terms from Joerg Arndt)

Cf. A007294, A179255 (nondecreasing differences), A179269, A320382, A320385.

Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences).

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
  }
  cnt
end
def A179254(n)
  (0..n).map{|i| f(i)}
end
p A179254(50) # Seiichi Manyama, Oct 12 2018

def A179254(n):
    has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
    allowed = lambda x: len(x) < 3 or has_increasing_diffs(x)
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011

A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169

Offset: 0

Seiichi Manyama, Oct 14 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(11) = 8 such partitions of 11:
01: [11]
02: [4, 7]
03: [5, 6]
04: [2, 4, 5]
05: [3, 4, 4]
06: [2, 3, 3, 3]
07: [1, 2, 2, 2, 2, 2]
08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(12) = 11 such partitions of 12:
01: [12]
02: [4, 8]
03: [5, 7]
04: [6, 6]
05: [2, 4, 6]
06: [3, 4, 5]
07: [4, 4, 4]
08: [3, 3, 3, 3]
09: [1, 2, 3, 3, 3]
10: [2, 2, 2, 2, 2, 2]
11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240026, A240027, A320466, A320470, A320510.
Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320509(n)
  (0..n).map{|i| f(i)}
end
p A320509(50)

A049989 a(n) is the number of arithmetic progressions of positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

1, 3, 6, 10, 14, 21, 26, 33, 42, 51, 58, 72, 80, 91, 107, 120, 130, 150, 161, 178, 199, 215, 228, 255, 272, 290, 316, 338, 354, 389, 406, 429, 460, 483, 508, 549, 569, 594, 630, 663, 685, 731, 754, 785, 833, 863, 888, 940, 969, 1007, 1054, 1090, 1118, 1175, 1212, 1253, 1305, 1342, 1373, 1444, 1476, 1515, 1577, 1621

Offset: 1

Clark Kimberling

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A047966, A049982, A049983, A049984, A049986, A049987, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

seq(n)={my(w=(sqrtint(8*n+1)-1)\2+1); Vec(x/(1-x)^2 + sum(k=2, n, x^k/(1 - if(k<=w, x^(k*(k-1)/2)))/(1-x^k) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Sep 28 2019

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049988(k). [Note that the offset of A049988 is 0.]
G.f.: (-1 + g.f. of A049988)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 28 2019

A325546 Number of compositions of n with weakly increasing differences.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 28, 41, 62, 87, 120, 170, 228, 303, 408, 534, 689, 899, 1145, 1449, 1842, 2306, 2863, 3571, 4398, 5386, 6610, 8039, 9716, 11775, 14157, 16938, 20293, 24166, 28643, 33995, 40134, 47199, 55540, 65088, 75994, 88776, 103328, 119886, 139126

Offset: 0

Gus Wiseman, May 10 2019

nonn

Also compositions of n whose plot is concave-up.
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (112)   (41)     (42)
                    (211)   (113)    (51)
                    (1111)  (212)    (114)
                            (311)    (123)
                            (1112)   (213)
                            (2111)   (222)
                            (11111)  (312)
                                     (321)
                                     (411)
                                     (1113)
                                     (2112)
                                     (3111)
                                     (11112)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A007294, A008965, A070211 (concave-down compositions), A173258, A175342, A240026, A325360, A325545, A325547, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A007294 (=breakdown by width).
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-w-1)\t + 1, v[i-w-(k-1)*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i/(1 - x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 28 2019

More terms from Alois P. Heinz, May 11 2019

A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 7, 6, 8, 10, 10, 11, 14, 13, 16, 19, 18, 20, 25, 23, 27, 31, 30, 34, 39, 37, 42, 48, 47, 50, 59, 56, 63, 70, 68, 74, 83, 82, 89, 97, 97, 104, 116, 113, 123, 133, 133, 142, 155, 153, 166, 178, 178, 189, 204, 204, 218, 232, 235, 247, 265, 265, 283, 299

Offset: 0

Seiichi Manyama, Oct 13 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are strictly decreasing. The Heinz numbers of these partitions are given by A325457. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 8 such partitions of 10:
01: [10]
02: [1, 9]
03: [2, 8]
04: [3, 7]
05: [4, 6]
06: [5, 5]
07: [1, 4, 5]
08: [2, 4, 4]
There are a(11) = 10 such partitions of 11:
01: [11]
02: [1, 10]
03: [2, 9]
04: [3, 8]
05: [4, 7]
06: [5, 6]
07: [1, 4, 6]
08: [1, 5, 5]
09: [2, 4, 5]
10: [3, 4, 4]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457.

Table[Length[Select[IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
  }
  cnt
end
def A320470(n)
  (0..n).map{|i| f(i)}
end
p A320470(50)

A179255 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 8, 9, 10, 13, 15, 16, 22, 24, 26, 33, 36, 39, 50, 54, 58, 70, 77, 83, 100, 109, 116, 137, 150, 159, 186, 202, 216, 249, 270, 288, 328, 355, 379, 428, 462, 491, 554, 597, 633, 707, 760, 807, 899, 964, 1020, 1127, 1211, 1282, 1412, 1512, 1596, 1750, 1873, 1976, 2160, 2305, 2434, 2652, 2826, 2978

Offset: 0

Joerg Arndt, Jan 05 2011

nonn

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.

			There are a(17) = 26 such partitions of 17:
01:  [ 1 2 3 4 7 ]
02:  [ 1 2 3 11 ]
03:  [ 1 2 4 10 ]  *
04:  [ 1 2 5 9 ]   *
05:  [ 1 2 14 ]   *
06:  [ 1 3 5 8 ]
07:  [ 1 3 13 ]   *
08:  [ 1 4 12 ]   *
09:  [ 1 5 11 ]   *
10:  [ 1 16 ]   *
11:  [ 2 3 4 8 ]
12:  [ 2 3 5 7 ]
13:  [ 2 3 12 ]   *
14:  [ 2 4 11 ]   *
15:  [ 2 5 10 ]   *
16:  [ 2 15 ]   *
17:  [ 3 4 10 ]   *
18:  [ 3 5 9 ]   *
19:  [ 3 14 ]   *
20:  [ 4 5 8 ]   *
21:  [ 4 13 ]   *
22:  [ 5 12 ]   *
23:  [ 6 11 ]   *
24:  [ 7 10 ]   *
25:  [ 8 9 ]   *
26:  [ 17 ]   *
The 21 partitions marked with * have strictly increasing differences, see the example for A179254.
- _Joerg Arndt_, Mar 31 2014

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000 (terms 0..241 from Joerg Arndt)

Cf. A009994.
Cf. A179254 (strictly increasing differences), A179269, A007294.
Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences), A320382.

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse
  }
  cnt
end
def A179255(n)
  (0..n).map{|i| f(i)}
end
p A179255(50) # Seiichi Manyama, Oct 12 2018

def A179255(n):
    has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0
    allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011

cp's OEIS Frontend

A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

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A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

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A325360 Heinz numbers of integer partitions whose differences are weakly increasing.

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A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

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A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.

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A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

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A049989 a(n) is the number of arithmetic progressions of positive integers, nondecreasing with sum <= n.

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A325546 Number of compositions of n with weakly increasing differences.

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