A225486 -id:A225486 - cp's OEIS frontend

A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 3, 0, 0, 1, 3, 4, 3, 0, 0, 0, 1, 1, 4, 8, 1, 0, 0, 0, 1, 3, 6, 9, 3, 0, 0, 0, 0, 1, 2, 8, 12, 7, 0, 0, 0, 0, 0, 1, 3, 11, 17, 10, 0, 0, 0, 0, 0, 0, 1, 1, 11, 26, 17, 0, 0, 0, 0, 0, 0, 0, 1, 5, 19, 25, 27

Offset: 0

Gus Wiseman, Apr 18 2019

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (this sequence).

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  3  0
  0  1  3  4  3  0  0
  0  1  1  4  8  1  0  0
  0  1  3  6  9  3  0  0  0
  0  1  2  8 12  7  0  0  0  0
  0  1  3 11 17 10  0  0  0  0  0
  0  1  1 11 26 17  0  0  0  0  0  0
  0  1  5 19 25 27  0  0  0  0  0  0  0
  0  1  1 17 44 38  0  0  0  0  0  0  0  0
  0  1  3 25 53 52  1  0  0  0  0  0  0  0  0
  0  1  3 29 63 76  4  0  0  0  0  0  0  0  0  0
  0  1  4 37 83 98  8  0  0  0  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)  (333)        (54)      (441)       (3321)
       (111111111)  (63)      (522)       (4221)
                    (72)      (711)       (4311)
                    (81)      (3222)      (5211)
                    (432)     (6111)      (32211)
                    (531)     (22221)     (42111)
                    (621)     (33111)     (321111)
                    (222111)  (51111)
                              (411111)
                              (2211111)
                              (3111111)
                              (21111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 2 is A032741. Column k = 3 is A325245.
Cf. A181819, A225486, A323014, A323023, A325254, A325258, A325277.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or this sequence (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==k&]],{n,0,16},{k,0,n}]

\\ depth(p) gives adjusted frequency depth of partition.
depth(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L; r++); r)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1+depth(Vec(p))]++); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A225485 Number of partitions of n that have frequency depth k, an array read by rows.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 4, 3, 1, 1, 4, 8, 1, 1, 3, 6, 9, 3, 1, 2, 8, 12, 7, 1, 3, 11, 17, 10, 1, 1, 11, 26, 17, 1, 5, 19, 25, 27, 1, 1, 17, 44, 38, 1, 3, 25, 53, 52, 1, 1, 3, 29, 63, 76, 4

Offset: 1

Clark Kimberling, May 08 2013

Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h).  Let f(i) be the frequency of y(i) in S.  Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m-1),S) for m>1.  Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.
Equivalently, the frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). - Gus Wiseman, Apr 19 2019
From Clark Kimberling, Sep 26 2023: (Start)
Below, m^n abbreviates the sum m+...+m of n terms. In the following list, the numbers p_1,...,p_k are distinct, m >= 1, and k >= 1. The forms of the partitions being counted are as follows:
   column 1: [n],
   column 2: [m^k],
   column 3: [p_1^m,...,p_k^m],
   column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.
Column 3 is of special interest. Assume first that m = 1, so that the form of partition being counted is p = [p_1,...,p_k], with conjugate given by [q_1,...,q_m] where  q_i is the number of parts of p that are >= i. Since the p_i are distinct, the distinct parts of q are the integers 1,2,...,k. For the general case that m >= 1, the distinct parts of q are the integers m,...,km. Let S(n) denote the set of partitions of n counted by column 3. Then if a and b are in the set S*(n) of conjugates of partitions in S(n), and if a > b, then a - b is also in S*(n). Call this the subtraction property. Conversely, if a partition q has the subtraction property, then q must consist of a set of numbers m,..,km for some m. Thus, column 3 counts the partitions of n that have the subtraction property. (End)

			The first 9 rows:
  n = 1 .... 0
  n = 2 .... 1..1
  n = 3 .... 1..1..1
  n = 4 .... 1..2..1..1
  n = 5 .... 1..1..2..3
  n = 6 .... 1..3..4..3
  n = 7 .... 1..1..4..8..1
  n = 8 .... 1..3..6..9..3
  n = 9 .... 1..2..8.12..7
For the 7 partitions of 5, successive frequencies are shown here:
  5 -> 1 (depth 1)
  41 -> 11 -> 2 -> 1 (depth 3)
  32 -> 11 -> 2 -> 1 (depth 3)
  311 -> 12 -> 11 -> 2 -> 1 (depth 4)
  221 -> 12 -> 11 -> 2 -> 1 (depth 4)
  2111 -> 13 -> 11 -> 2 -> 1 (depth 4)
  11111 -> 5 -> 1 (depth 2)
Summary: 1 partition has depth 1; 1 has depth 2; 2 have 3; and 3 have 4, so that the row for n = 5 is 1..1..2..3 .

Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Clark Kimberling)

Row sums are A000041.
Column k = 2 is A032741.
Column k = 3 is A325245.
a(n!) = A325272(n).
Cf. A181819, A182850, A225486, A323014, A323023, A325238, A325239, A325254.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], i]],
  {i, 1, Length[IntegerPartitions[n]]}];
Flatten[Table[Count[u[n], k], {n, 2, 25}, {k, 1, Max[u[n]]}]]

A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, -1, 0, 1, 3, 2, 1, 1, 2, 5, -4, -2, -1, 0, 2, 7, 9, 5, 3, 2, 2, 4, 11, -21, -12, -7, -4, -2, 0, 4, 15, 49, 28, 16, 9, 5, 3, 3, 7, 22, -112, -63, -35, -19, -10, -5, -2, 1, 8, 30, 249, 137, 74, 39, 20, 10, 5, 3, 4, 12, 42, -539, -290, -153, -79, -40, -20, -10, -5, -2, 2, 14, 56

Offset: 0

Alois P. Heinz, Dec 04 2010

Odlyzko showed that the k-th differences of A000041(n) alternate in sign with increasing n up to a certain index n_0(k) and then stay positive.

Are there any zeros after the first four, which all lie in columns k = 1, 2? - Gus Wiseman, Dec 15 2024

			Square array A(n,k) begins:
   1,  0,  1, -1,  2,  -4,   9,  ...
   1,  1,  0,  1, -2,   5, -12,  ...
   2,  1,  1, -1,  3,  -7,  16,  ...
   3,  2,  0,  2, -4,   9, -19,  ...
   5,  2,  2, -2,  5, -10,  20,  ...
   7,  4,  0,  3, -5,  10, -20,  ...
  11,  4,  3, -2,  5, -10,  22,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Charles Knessl, Asymptotic Behavior of High-Order Differences of the Partition Function, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), 237-254.

Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.
Main diagonal gives A379378.
Cf. A119712, A155861.
For primes we have A095195 or A376682.
Row n = 0 is A281425.
Row n = 1 is A320590 except first term.
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Antidiagonal sums are A377056, absolute value version A378621.
The version for strict partitions is A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.

A41:= combinat[numbpart]:
DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
A:= (n,k)-> (DD@@k)(A41)(n):
seq(seq(A(n, d-n), n=0..d), d=0..11);

max = 11; a41 = Array[PartitionsP, max+1, 0]; a[n_, k_] := Differences[a41, k][[n+1]]; Table[a[n, k-n], {k, 0, max}, {n, 0, k}] // Flatten (* Jean-François Alcover, Aug 29 2014 *)
nn=5;Table[Table[Sum[(-1)^(k-i)*Binomial[k,i]*PartitionsP[n+i],{i,0,k}],{k,0,nn}],{n,0,nn}] (* Gus Wiseman, Dec 15 2024 *)

A(n,k) = (Delta^(k) A000041)(n).

A(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A000041(n+i). In words, row x is the inverse zero-based binomial transform of A000041 shifted left x times. - Gus Wiseman, Dec 15 2024

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

Original entry on oeis.org

1, 0, 1, -1, 2, -4, 9, -21, 49, -112, 249, -539, 1143, -2396, 5013, -10550, 22420, -48086, 103703, -223806, 481388, -1029507, 2187944, -4625058, 9742223, -20490753, 43111808, -90840465, 191773014, -405523635, 858378825, -1817304609, 3845492204, -8129023694, 17162802918, -36191083386

Offset: 0

Ilya Gutkovskiy, Oct 05 2017

sign

a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...

Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - Gus Wiseman, Feb 06 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
A. M. Odlyzko, Differences of the partition function, Acta Arithmetica 49.3 (1988): 237-254.
Dennis Stanton and Doron Zeilberger, The Odlyzko conjecture and O’Hara’s unimodality proof, Proceedings of the American Mathematical Society 107.1 (1989): 39-42.

Cf. A128566, A292463, A292541, A292613.
Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
Cf. A218481, A294466, A095051.
Cf. A266232, A294467, A293467, A294468.
Row n=0 of A175804 (row n=1 is A320590 except first term).
Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.

b:= proc(n, k) option remember; `if`(k=0,
      combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2024

Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 06 2017 *)

a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - Vaclav Kotesovec, Oct 06 2017
a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325755.

			The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

A325705 Number of integer partitions of n containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 4, 3, 7, 8, 16, 15, 24, 28, 39, 44, 68, 80, 98, 130, 167, 200, 259, 320, 396, 497, 601, 737, 910, 1107, 1335, 1631, 1983, 2372, 2887, 3439, 4166, 4949, 5940, 7043, 8450, 9980, 11884, 13984, 16679, 19493, 23162, 27050, 31937, 37334, 43926

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325706.

			The partition (4,2,1,1,1,1) has distinct multiplicities {1,4}, both of which belong to the partition, so it is counted under a(10).
The a(0) = 1 through a(10) = 16 partitions:
  ()  (1)  (21)  (22)   (41)   (51)    (61)    (71)     (81)     (91)
                 (31)   (221)  (321)   (421)   (431)    (333)    (541)
                 (211)         (2211)  (3211)  (521)    (531)    (631)
                               (3111)          (3221)   (621)    (721)
                                               (4211)   (3321)   (3322)
                                               (32111)  (4221)   (3331)
                                               (41111)  (5211)   (4321)
                                                        (32211)  (5221)
                                                                 (6211)
                                                                 (32221)
                                                                 (33211)
                                                                 (42211)
                                                                 (43111)
                                                                 (322111)
                                                                 (421111)
                                                                 (511111)

Cf. A109297, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325706, A325707, A325755.

Table[Length[Select[IntegerPartitions[n],SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A325282 Maximum adjusted frequency depth among integer partitions of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A325258 a(1) = 1; otherwise, first differences of Levine's sequence A011784.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 28, 171, 2624, 172613, 139584150, 6837485347187, 266437138079023501057, 508009471379222384299345337895696, 37745517525533091954228691786161750063795478326636142, 5347426383812697233786139576220412396732847744407175515852823296919414647252347610750

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

a(n) is the number of nonnegative integers k such that the maximum adjusted frequency depth among integer partitions of k is n. For example, the a(5) = 7 numbers are 7, 8, 9, 10, 11, 12, and 13.

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n). The maximum adjusted frequency depth for partitions of n is A325282(n).

Run lengths of A325282.

Cf. A011784, A181819, A181821, A182850, A182857, A225485, A225486, A323014, A325238, A325254, A325278, A325280, A325283.

grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
ReplacePart[Differences[Last/@NestList[grw,{1,1},9]],2->1]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

cp's OEIS Frontend

A325280 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.

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A225485 Number of partitions of n that have frequency depth k, an array read by rows.

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A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

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A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

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A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

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A325705 Number of integer partitions of n containing all of their distinct multiplicities.

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A325282 Maximum adjusted frequency depth among integer partitions of n.

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A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

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