A325356 -id:A325356 - cp's OEIS frontend

A007294 Number of partitions of n into nonzero triangular numbers.

1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.
a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.
Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters, Apr 08 2008
Also partitions with condition as in A179255, and additionally, if more than one part, first difference >= first part: for example, a(10)=7 as there are 7 such partitions of 10:  1+2+3+4 = 1+2+7 = 1+3+6 = 1+9 = 2+8 = 3+7 = 10. - Joerg Arndt, Mar 22 2011
Number of members of A181818 with a bigomega value of n (cf. A001222). - Matthew Vandermast, May 19 2012

			6 = 3+3 = 3+1+1+1 = 1+1+1+1+1+1 so a(6) = 4.
a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.
From _Gus Wiseman_, May 03 2019: (Start)
The a(1) = 1 through a(9) = 6 partitions using nonzero triangular numbers are the following. The Heinz numbers of these partitions are given by A325363.
  1   11   3     31     311     6        61        611        63
           111   1111   11111   33       331       3311       333
                                3111     31111     311111     6111
                                111111   1111111   11111111   33111
                                                              3111111
                                                              111111111
The a(1) = 1 through a(10) = 7 partitions with weakly decreasing multiplicities are the following. Equivalent to Matthew Vandermast's comment, the Heinz numbers of these partitions are given by A025487 (products of primorial numbers).
  1  11  21   211   2111   321     3211     32111     32211      4321
         111  1111  11111  2211    22111    221111    222111     322111
                           21111   211111   2111111   321111     2221111
                           111111  1111111  11111111  2211111    3211111
                                                      21111111   22111111
                                                      111111111  211111111
                                                                 1111111111
The a(1) = 1 through a(11) = 7 partitions with weakly increasing differences (where the last part is taken to be zero) are the following. The Heinz numbers of these partitions are given by A325362 (A = 10, B = 11).
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)     (B)
            (21)  (31)  (41)  (42)   (52)   (62)   (63)   (73)    (83)
                              (51)   (61)   (71)   (72)   (82)    (92)
                              (321)  (421)  (521)  (81)   (91)    (A1)
                                                   (531)  (631)   (731)
                                                   (621)  (721)   (821)
                                                          (4321)  (5321)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006.
G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), p. 191.
Jorge A. Campos-Gonzalez-Angulo, Raphael F. Ribeiro, and Joel Yuen-Zhou, Generalization of the Tavis-Cummings model for multi-level anharmonic systems, arXiv:2101.09475 [physics.optics], 2021.
Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, Journal of Integer Sequences, Vol. 7, 2004.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000217, A051533, A000294.
Cf. A102462.
Row sums of array A176723 and triangle A176724. - Wolfdieter Lang, Jul 19 2010
Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of nondecreasing). - Joerg Arndt, Mar 22 2011
Cf. A024940, A280366.
Row sums of A319797.
Cf. A007862, A025487, A240026, A320509, A325324, A325354, A325356, A325362, A325363.

a007294 = p $ tail a000217_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 28 2013

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-i*(i+1)/2, i)
      fi
    end:
a:= n-> b(n, floor(sqrt(2*n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 22 2011
isNondecrP :=proc(L) slp := DIFF(DIFF(L)) ; min(op(%)) >= 0 ; end proc:
A007294 := proc(n) local a, p; a := 0 ; if n = 0 then return 1 ; end if; for p in combinat[partition](n) do if nops(p) = nops(convert(p, set)) then if isNondecrP(p) then if nops(p) =1 then a := a+1 ; elif op(2, p) >= 2*op(1, p) then a := a+1; end if; end if; end if; end do; a ; end proc:
seq(A007294(n), n=0..30) ; # R. J. Mathar, Jan 07 2011

CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i == 0, 0, True, b[n, i-1]+b[n-i*(i+1)/2, i]]; a[n_] := b[n, Floor[Sqrt[2*n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[Append[#,0]]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)
nmax = 58; t = Table[PolygonalNumber[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[t, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

N=66; Vec(1/prod(k=1,N,1-x^(k*(k+1)\2))+O(x^N)) \\ Joerg Arndt, Apr 14 2013

from functools import lru_cache
from sympy import divisors
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A007294(n):
    @lru_cache(maxsize=None)
    def a(n): return is_square((n<<3)+1)
    @lru_cache(maxsize=None)
    def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
    return (c(n)+sum(c(k)*A007294(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

def A007294(n):
    has_nondecreasing_diffs = lambda x: min(differences(x, 2)) >= 0
    special = lambda x: (x[1]-x[0]) >= x[0]
    allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_nondecreasing_diffs(x))
    return len([1 for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011

G.f.: 1/Product_{k>=2} (1-z^binomial(k, 2)).
For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller, Aug 26 2003
For n>0, a(n) is Euler Transform of [1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...], i.e A010054, n>0. - Benedict W. J. Irwin, Jul 29 2016
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2) / (2^(7/2) * sqrt(3) * Pi * n^(3/2)) [Brigham 1950 (exponential part), Almkvist 2006]. - Vaclav Kotesovec, Dec 31 2016
G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i} (1 - x^(j*(j+1)/2)). - Ilya Gutkovskiy, May 07 2017

Additional comments from Roland Bacher, Jun 17 2001

A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419

Offset: 0

Joerg Arndt, Mar 31 2014

nonn

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 weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

			There are a(10) = 27 such partitions of 10:
01:  [ 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 1 3 ]
04:  [ 1 1 1 1 1 1 4 ]
05:  [ 1 1 1 1 1 2 3 ]
06:  [ 1 1 1 1 1 5 ]
07:  [ 1 1 1 1 2 4 ]
08:  [ 1 1 1 1 6 ]
09:  [ 1 1 1 2 5 ]
10:  [ 1 1 1 7 ]
11:  [ 1 1 2 6 ]
12:  [ 1 1 3 5 ]
13:  [ 1 1 8 ]
14:  [ 1 2 3 4 ]
15:  [ 1 2 7 ]
16:  [ 1 3 6 ]
17:  [ 1 9 ]
18:  [ 2 2 2 2 2 ]
19:  [ 2 2 2 4 ]
20:  [ 2 2 6 ]
21:  [ 2 3 5 ]
22:  [ 2 8 ]
23:  [ 3 3 4 ]
24:  [ 3 7 ]
25:  [ 4 6 ]
26:  [ 5 5 ]
27:  [ 10 ]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..203 from Joerg Arndt)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A240027 (strictly increasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.

Table[Length[Select[IntegerPartitions[n],OrderedQ[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.reverse
  }
  cnt
end
def A240026(n)
  (0..n).map{|i| f(i)}
end
p A240026(50) # Seiichi Manyama, Oct 13 2018

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.

Original entry on oeis.org

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

Offset: 2

Alexander Adamchuk, Apr 27 2007

nonn

The indices k of the first appearance of number n in a(k) are listed in A063778(n) = {2,3,6,15,36,225,...} = Least number k>1 such that k could be represented in n different ways as general m-gonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)).
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n whose augmented differences are all equal, where the augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k; for example aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Equivalently, a(n) is the number of integer partitions of n whose differences are all equal to the last part minus one. The Heinz numbers of these partitions are given by A307824. For example, the a(35) = 5 partitions are:
  (35)
  (23,12)
  (11,9,7,5,3)
  (8,7,6,5,4,3,2)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
(End)

			a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2).

Alois P. Heinz, Table of n, a(n) for n = 2..10000
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A063778, A177025.
Column k=0 of A239550.
Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459.

A129654 := proc(n) local resul, dvs, i, r, m ;
   dvs := numtheory[divisors](2*n) ;
   resul := 0 ;
   for i from 1 to nops(dvs) do
      r := op(i, dvs) ;
      if r > 1 then
         m := (2*n/r-4+2*r)/(r-1) ;
         if is(m, integer) then
            resul := resul+1 ;
         fi ;
      fi ;
   od ;
   RETURN(resul) ;
end: # R. J. Mathar, May 14 2007

a[n_] := (dvs = Divisors[2*n]; resul = 0; For[i = 1, i <= Length[dvs], i++, r = dvs[[i]]; If[r > 1, m = (2*n/r-4+2*r)/(r-1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Sep 13 2012, translated from R. J. Mathar's Maple program *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* Jonathan Sondow, May 09 2014 *)
atpms[n_]:=Select[Join@@Table[i*Range[k,1,-1],{k,n},{i,0,n}],Total[#+1]==n&];
Table[Length[atpms[n]],{n,100}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ Daniel Suteu, Dec 22 2018

a(n) = A177025(n) + 1.

G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020

A325394 Heinz numbers of integer partitions whose augmented differences are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167

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 augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325356.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

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

Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 19, 24, 25, 36, 37, 43, 54, 63, 64, 80, 81, 100, 113, 122, 123, 151, 166, 178, 195, 217, 218, 269, 270, 295, 316, 332, 372, 424, 425, 447, 472, 547, 550, 616, 617, 659, 750, 777, 782, 862, 885, 995, 1032, 1083, 1090, 1176, 1275

Offset: 0

Gus Wiseman, May 02 2019

nonn

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).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325400.

			The a(1) = 1 through a(8) = 15 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (112)   (113)    (33)      (34)       (35)
                    (1111)  (1112)   (114)     (115)      (44)
                            (11111)  (123)     (124)      (116)
                                     (222)     (223)      (125)
                                     (1113)    (1114)     (224)
                                     (11112)   (11113)    (1115)
                                     (111111)  (111112)   (1124)
                                               (1111111)  (2222)
                                                          (11114)
                                                          (111113)
                                                          (1111112)
                                                          (11111111)

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

Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

Table[Length[Select[Sort/@IntegerPartitions[n],And@@Table[OrderedQ[Differences[#,k]],{k,0,Length[#]}]&]],{n,0,30}]

A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 5, 5, 4, 5, 6, 5, 7, 7, 7, 7, 9, 7, 10, 10, 8, 11, 13, 10, 13, 14, 12, 14, 17, 13, 17, 19, 17, 18, 22, 19, 22, 24, 21, 24, 28, 24, 29, 30, 28, 31, 35, 30, 35, 40, 36

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325395.

			The a(28) = 10 partitions:
  (28)
  (18,10)
  (17,11)
  (16,12)
  (15,13)
  (14,14)
  (12,10,6)
  (11,10,7)
  (10,10,8)
  (8,8,7,5)
For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000

Cf. A000837, A007294, A049988, A098859, A325351, A325356, A325360, A325391, A325395.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 11, 13, 14, 15, 18, 20, 21, 24, 26, 28, 33, 36, 38, 43, 46, 49, 56, 60, 63, 71, 76, 80, 90, 96, 100, 112, 120, 125, 139, 149, 155, 171, 183, 190, 208, 223, 232, 252, 269, 280, 304, 325, 338, 364, 387, 403

Offset: 0

Gus Wiseman, May 01 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325396.

			The a(1) = 1 through a(11) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
            (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
                              (51)  (61)   (71)   (72)   (82)   (92)
                                    (421)  (521)  (81)   (91)   (101)
                                                  (621)  (631)  (731)
                                                         (721)  (821)

Fausto A. C. Cariboni, 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. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325389.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (52)       (53)
             (111)  (211)   (41)     (51)      (61)       (62)
                    (1111)  (311)    (321)     (421)      (71)
                            (2111)   (411)     (511)      (521)
                            (11111)  (3111)    (3211)     (611)
                                     (21111)   (4111)     (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Mathematica
```
aug[y_]:=Table[If[i
				
```

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

Original entry on oeis.org

4, 8, 15, 16, 32, 55, 64, 90, 105, 119, 128, 225, 253, 256, 403, 512, 540, 550, 697, 893, 935, 1024, 1155, 1350, 1357, 1666, 1943, 2048, 2263, 3025, 3071, 3150, 3240, 3375, 3451, 3927, 3977, 4096, 4429, 5123, 5500, 5566, 6731, 7735, 8083, 8100, 8192, 9089

Offset: 1

Gus Wiseman, Nov 06 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A sequence is periodic if its cyclic rotations are not all different.

			The sequence of terms together with their augmented differences of prime indices begins:
     4: (1,1)
     8: (1,1,1)
    15: (2,2)
    16: (1,1,1,1)
    32: (1,1,1,1,1)
    55: (3,3)
    64: (1,1,1,1,1,1)
    90: (2,1,2,1)
   105: (2,2,2)
   119: (4,4)
   128: (1,1,1,1,1,1,1)
   225: (1,2,1,2)
   253: (5,5)
   256: (1,1,1,1,1,1,1,1)
   403: (6,6)
   512: (1,1,1,1,1,1,1,1,1)
   540: (2,1,1,2,1,1)
   550: (3,1,3,1)
   697: (7,7)
   893: (8,8)

Complement of A329133.
These are the Heinz numbers of the partitions counted by A329143.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Numbers whose differences of prime indices are periodic are A329134.
Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

A329136 Number of integer partitions of n whose augmented differences are an aperiodic word.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 10, 14, 19, 28, 40, 53, 75, 99, 131, 172, 226, 294, 380, 488, 617, 787, 996, 1250, 1565, 1953, 2425, 3003, 3705, 4559, 5589, 6836, 8329, 10132, 12292, 14871, 17950, 21629, 25988, 31169, 37306, 44569, 53139, 63247, 75133, 89111, 105515, 124737

Offset: 0

Gus Wiseman, Nov 09 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

A sequence is aperiodic if its cyclic rotations are all different.

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (2,2)    (4,1)      (3,3)        (4,3)
                   (3,1)    (2,2,1)    (4,2)        (5,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (2,2,2)      (3,2,2)
                                       (3,2,1)      (3,3,1)
                                       (4,1,1)      (4,2,1)
                                       (2,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (2,2,2,1)
                                       (2,1,1,1,1)  (3,2,1,1)
                                                    (4,1,1,1)
                                                    (2,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)
With augmented differences:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (1,2)    (4,1)      (1,3)        (2,3)
                   (3,1)    (1,2,1)    (3,2)        (4,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (1,1,2)      (1,3,1)
                                       (2,2,1)      (2,1,2)
                                       (4,1,1)      (3,2,1)
                                       (1,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (1,1,2,1)
                                       (2,1,1,1,1)  (2,2,1,1)
                                                    (4,1,1,1)
                                                    (1,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)

The Heinz numbers of these partitions are given by A329133.
The periodic version is A329143.
The non-augmented version is A329137.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose differences of prime indices are aperiodic are A329135.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325351, A325356, A329132, A329134, A329139, A329140.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

a(n) + A329143(n) = A000041(n).

cp's OEIS Frontend

A007294 Number of partitions of n into nonzero triangular numbers.

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A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

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A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m,r>1.

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A325394 Heinz numbers of integer partitions whose augmented differences are weakly increasing.

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A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

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A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

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A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.