A066186 -id:A066186 - cp's OEIS frontend

A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

0, 2, 3, 8, 10, 24, 28, 56, 72, 120, 154, 252, 312, 476, 615, 880, 1122, 1584, 1995, 2740, 3465, 4620, 5819, 7680, 9575, 12428, 15498, 19824, 24563, 31170, 38378, 48224, 59202, 73678, 90055, 111384, 135420, 166364, 201630, 246120, 297045, 360822

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Sum of all parts > 1 of the last section of the set of partitions of n.
Row sums of triangle A182710. Also row sums of other similar triangles as A138136 and A182711.
Partial sums give A194552. - Omar E. Pol, Sep 23 2013

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A066186, A133041, A138135, A138136, A138137, A138138, A138151, A138879, A139100.

Table[Total[Flatten[Select[IntegerPartitions[n],FreeQ[#,1]&]]],{n,50}] (* Harvey P. Dale, May 24 2015 *)
a[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = A002865(n)*n = (A000041(n) - A000041(n-1))*n = A138879(n) - A000041(n-1).
a(n) ~ Pi^2/6*A000070(n-2). - Peter Bala, Dec 23 2013
G.f.: x*f'(x), where f(x) = Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Jul 06 2019

Better definition from Omar E. Pol, Sep 23 2013

A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 6, 9, 12, 0, 1, 10, 17, 20, 20, 0, 1, 18, 39, 44, 35, 35, 0, 1, 34, 101, 122, 87, 66, 54, 0, 1, 66, 279, 392, 287, 180, 105, 86, 0, 1, 130, 797, 1370, 1119, 660, 311, 176, 128, 0, 1, 258, 2319, 5024, 4775, 2904, 1281, 558, 270, 192

Offset: 0

Alois P. Heinz, Feb 28 2013

In general, if k > 0 then column k is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - Vaclav Kotesovec, May 27 2018

			Square array A(n,k) begins:
:   0,  0,   0,   0,    0,     0,     0, ...
:   1,  1,   1,   1,    1,     1,     1, ...
:   3,  4,   6,  10,   18,    34,    66, ...
:   6,  9,  17,  39,  101,   279,   797, ...
:  12, 20,  44, 122,  392,  1370,  5024, ...
:  20, 35,  87, 287, 1119,  4775, 21447, ...
:  35, 66, 180, 660, 2904, 14196, 73920, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A006128, A066186, A066183, A229325, A229326, A229327, A229328, A229329, A229330, A229331, A229332.
Rows n=0-10 give: A000004, A000012, A052548, A229354, A229355, A229356, A229357, A229358, A229359, A229360, A229361.
Main diagonal gives A252761.
Cf. A213180.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
          (b(n-p*m, p-1, k)), m=0..n/p)))
    end:
A:= (n, k)-> b(n, n, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 0, l, l + {0, First[l]*p^k*m}]][b[n - p*m, p - 1, k]], { m, 0, n/p}]]] ; a[n_, k_] := b[n, n, k][[2]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; A[n_, k_] := Sum[T[n, j]*j^k, {j, 1, n}]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 15 2016 *)

A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.

A343341 Number of integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 28, 36, 58, 79, 111, 149, 209, 270, 368, 472, 618, 793, 1030, 1292, 1653, 2073, 2608, 3241, 4051, 4982, 6176, 7566, 9285, 11320, 13805, 16709, 20275, 24454, 29477, 35380, 42472, 50741, 60648, 72199, 85887, 101906, 120816

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of integer partitions of n that are either empty, or have greatest part not divisible by all the others.

			The a(5) = 1 through a(10) = 16 partitions:
  (32)  (321)  (43)    (53)     (54)      (64)
               (52)    (332)    (72)      (73)
               (322)   (431)    (432)     (433)
               (3211)  (521)    (522)     (532)
                       (3221)   (531)     (541)
                       (32111)  (3222)    (721)
                                (3321)    (3322)
                                (4311)    (4321)
                                (5211)    (5221)
                                (32211)   (5311)
                                (321111)  (32221)
                                          (33211)
                                          (43111)
                                          (52111)
                                          (322111)
                                          (3211111)

The complement is counted by A130689.
The dual version is A338470.
The Heinz numbers of these partitions are A343337.
The strict case is A343377.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
Cf. A066186, A083710, A083711, A097986, A098965, A341450, A343342, A343345, A343346, A343381, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

Original entry on oeis.org

1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470

Offset: 0

Omar E. Pol, Jul 29 2013

nonn

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

			For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15 |_ _ _ _      |
4 + 3                              |_ _ _ _|_    |
5 + 2                              |_ _ _    |   |
3 + 2 + 2                          |_ _ _|_ _|_  |
6 + 1                           11 |_ _ _      | |
3 + 3 + 1                          |_ _ _|_    | |
4 + 2 + 1                          |_ _    |   | |
2 + 2 + 2 + 1                      |_ _|_ _|_  | |
5 + 1 + 1                        7 |_ _ _    | | |
3 + 2 + 1 + 1                      |_ _ _|_  | | |
4 + 1 + 1 + 1                    5 |_ _    | | | |
2 + 2 + 1 + 1 + 1                  |_ _|_  | | | |
3 + 1 + 1 + 1 + 1                3 |_ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_|
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                        |_ _ _      |
.                                        |_ _ _|_    |
.                                        |_ _    |   |
.                             _ _ _ _ _  |_ _|_ _|_  |
.                            |_ _ _    | |_ _ _    | |
.                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | |
.                  |_ _    | |_ _    | | |_ _    | | |
.           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | |
.     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | |
. _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4    10     18       33         52          87

Omar E. Pol, Illustration of the seven regions of 5
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

A339106 Triangle read by rows: T(n,k) = A000203(n-k+1)*A000041(k-1), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 4, 3, 2, 7, 4, 6, 3, 6, 7, 8, 9, 5, 12, 6, 14, 12, 15, 7, 8, 12, 12, 21, 20, 21, 11, 15, 8, 24, 18, 35, 28, 33, 15, 13, 15, 16, 36, 30, 49, 44, 45, 22, 18, 13, 30, 24, 60, 42, 77, 60, 66, 30, 12, 18, 26, 45, 40, 84, 66, 105, 88, 90, 42, 28, 12, 36, 39, 75, 56, 132, 90, 154, 120, 126, 56

Offset: 1

Omar E. Pol, Nov 23 2020

Conjecture 1: T(n,k) is the sum of all divisors of all (n - k + 1)'s in the n-th row of triangle A176206, assuming that A176206 has offset 1. The same for the triangle A340061.

Conjecture 2: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

			Triangle begins:
   1;
   3,  1;
   4,  3,  2;
   7,  4,  6,  3;
   6,  7,  8,  9,  5;
  12,  6, 14, 12, 15,  7;
   8, 12, 12, 21, 20, 21,  11;
  15,  8, 24, 18, 35, 28,  33,  15;
  13, 15, 16, 36, 30, 49,  44,  45,  22;
  18, 13, 30, 24, 60, 42,  77,  60,  66,  30;
  12, 18, 26, 45, 40, 84,  66, 105,  88,  90,  42;
  28, 12, 36, 39, 75, 56, 132,  90, 154, 120, 126, 56;
...
For n = 6 the calculation of every term of row 6 is as follows:
-------------------------
k   A000041        T(6,k)
1      1  *  12  =   12
2      1  *  6   =    6
3      2  *  7   =   14
4      3  *  4   =   12
5      5  *  3   =   15
6      7  *  1   =    7
.         A000203
-------------------------
The sum of row 6 is 12 + 6 + 14 + 12 + 15 + 7 = 66, equaling A066186(6).

Mirror of A221529.
Row sums give A066186 (conjectured).
Main diagonal gives A000041.
Columns 1 and 2 give A000203.
Column 3 gives A074400.
Column 4 gives A272027.
Column 5 gives A274535.
Column 6 gives A319527.
Cf. A176206, A340061.

T[n_, k_] := DivisorSigma[1, n - k + 1] * PartitionsP[k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jan 08 2021 *)

T(n, k) = sigma(n-k+1)*numbpart(k-1); \\ Michel Marcus, Jan 08 2021

T(n,k) = sigma(n-k+1)*p(k-1), n >= 1, 1 <= k <= n.

A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 0, 3, 3, 4, 0, 4, 5, 2, 3, 0, 5, 7, 8, 6, 4, 0, 6, 11, 6, 6, 4, 5, 0, 7, 15, 16, 9, 12, 5, 6, 0, 8, 22, 14, 18, 8, 10, 6, 7, 0, 9, 30, 30, 18, 20, 15, 12, 7, 8, 0, 10, 42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11, 56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12

Offset: 1

Omar E. Pol, Feb 24 2012

For further properties of this triangle see also A182703.

			Triangle begins:
   1;
   1,  2;
   2,  0,  3;
   3,  4,  0,  4;
   5,  2,  3,  0,  5;
   7,  8,  6,  4,  0,  6;
  11,  6,  6,  4,  5,  0,  7;
  15, 16,  9, 12,  5,  6,  0,  8;
  22, 14, 18,  8, 10,  6,  7,  0,  9;
  30, 30, 18, 20, 15, 12,  7,  8,  0, 10;
  42, 30, 30, 20, 20, 12, 14,  8,  9,  0, 11;
  56, 54, 42, 40, 25, 30, 14, 16,  9, 10,  0, 12;
...
From _Omar E. Pol_, Nov 28 2020: (Start)
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of triangle A182703
.      * * * * * * *
.      1,2,3,4,5,6,7 --> Row 7 of triangle A002260
.      = = = = = = =
.     11,6,6,4,5,0,7 --> Row 7 of this triangle
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)

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

Column 1 is A000041.
Leading diagonal gives A000027.
Second diagonal gives A000007.
Row sums give A138879.
Cf. A002260, A066186, A135010, A138121, A138785, A138880, A182703, A194812, A207031.

T(n,k) = k*A182703(n,k).

A337209 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000203, the sum of divisors function.

Original entry on oeis.org

1, 3, 1, 4, 3, 1, 1, 7, 4, 3, 3, 1, 1, 1, 6, 7, 4, 4, 3, 3, 3, 1, 1, 1, 1, 1, 12, 6, 7, 7, 4, 4, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 8, 12, 6, 6, 7, 7, 7, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 8, 12, 12, 6, 6, 6, 7, 7, 7, 7, 7, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Omar E. Pol, Nov 27 2020

Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

			Triangle begins:
   1;
   3,  1;
   4,  3, 1, 1;
   7,  4, 3, 3, 1, 1, 1;
   6,  7, 4, 4, 3, 3, 3, 1, 1, 1, 1, 1;
  12,  6, 7, 7, 4, 4, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
   8, 12, 6, 6, 7, 7, 7, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Sum of divisors of terms of A176206.
Cf. A339278 (another version).
Cf. A000070, A000203, A066186, A221529, A238442, A339258.

A337209row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A337209row,10] (* Paolo Xausa, Sep 02 2023 *)

f(n) = sum(k=0, n-1, numbpart(k));
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); );} \\ Michel Marcus, Jan 13 2021

T(n,k) = A000203(A176206(n,k)).

A007870 Determinant of character table of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 6, 96, 2880, 9953280, 100329062400, 10651768002183168000, 150283391703941024789299200000, 9263795272057860957392207640004657152000000000, 16027108137650009941734148595388542471170145479274004480000000000000

Offset: 0

Peter J. Cameron, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]

nonn

			1 + x + 2*x^2 + 6*x^3 + 96*x^4 + 2880*x^5 + 9953280*x^6 + 100329062400*x^7 + ...
The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)} with product 4*3*1*2*2*2*1*1*1*1*1*1 = 96. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..18
Amritanshu Prasad, Symmetric Functions, Chapter 5, Representation Theory: a Combinatorial Viewpoint, Cambridge Studies in Adv. Math. 147 (2014), p. 107.
F. W. Schmidt and R. Simion, On a partition identity, J. Combin. Theory, A 36 (1984), 249-252.
D. Vaintrob, A product identity for partitions, MathOverflow, June 2012.

Row-products of A302246 and A302247.
Cf. A000041, A000142, A006128, A006906, A066186, A066633, A086644, A325501, A325504, A325507, A325536.
Cf. A063073.

List(List([0..11],n->Flat(Partitions(n))),Product); # Muniru A Asiru, Dec 21 2018

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1$2], ((f, g)->
      [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 30 2013

Needs["Combinatorica`"]; Table[Times@@Flatten[Partitions[n]], {n, 10}]
a[ n_] := If[n < 0, 0, Times @@ Flatten @ IntegerPartitions @ n] (* Michael Somos, Jun 11 2012 *)
Table[Exp[Total[Map[Log, IntegerPartitions [n]], 2]], {n, 1, 25}] (* Richard R. Forberg, Dec 08 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 1}, Function[{f, g}, {f[[1]] + g[[1]], f[[2]]*g[[2]]*i^g[[1]]}][If[i < 2, {0, 1}, b[n, i - 1]], If[i > n, {0, 1}, b[n - i, i]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

from sympy import prod
from sympy.utilities.iterables import ordered_partitions
a = lambda n: prod(map(prod, ordered_partitions(n))) if n > 0 else 1
print([a(n) for n in range(0, 12)]) # Darío Clavijo, Feb 22 2024

Product of all parts of all partitions of n.
From Gus Wiseman, May 09 2019: (Start)
a(n) = A003963(A325501(n)).
A001222(a(n)) = A325536(n).
A001221(a(n)) = A000720(n).
(End)

A302246 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 05 2018

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nonincreasing order. In other words: row n lists in nonincreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

			Triangle begins:
  1;
  2,1,1;
  3,2,1,1,1,1;
  4,3,2,2,2,1,1,1,1,1,1,1;
  5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1;
  6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There is only one 4, only one 3, three 2's and seven 1's, so the 4th row of this triangle is [4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nonincreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Paolo Xausa, Table of n, a(n) for n = 1..9687, (rows 1..18 of triangle, flattened).

Both column 1 and 2 are A000027.
Row n has length A006128(n).
The sum of row n is A066186(n).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
First differs from A036037, A080577, A181317, A237982 and A239512 at a(13) = T(4,3).
Cf. A302247 (mirror).
Cf. A000041, A027550, A176206, A221529, A336812, A338156.

nrows=10;Array[ReverseSort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)

row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list), , 4); \\ Michel Marcus, Jun 16 2022

A302247 Irregular triangle read by rows in which row n lists all parts of all partitions of n, in nondecreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 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

Offset: 1

Omar E. Pol, Apr 05 2018

Also due to the correspondence divisor/part row n lists the terms of the n-th row of A338156 in nondecreasing order. In other words: row n lists in nondecreasing order the divisors of the terms of the n-th row of A176206. - Omar E. Pol, Jun 16 2022

			Triangle begins:
  1;
  1,1,2;
  1,1,1,1,2,3;
  1,1,1,1,1,1,1,2,2,2,3,4;
  1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
  ...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. There are seven 1's, three 2's, only one 3 and only one 4, so the 4th row of this triangle is [1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4].
On the other hand for n = 4 the 4th row of A176206 is [4, 3, 2, 2, 1, 1, 1] and the divisors of these terms are [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1], [1] the same as the 4th row of A338156. These divisors listed in nondecreasing order give the 4th row of this triangle. - _Omar E. Pol_, Jun 16 2022

Paolo Xausa, Table of n, a(n) for n = 1..9687, (rows 1..18 of triangle, flattened)

Mirror of A302246.
Row n has length A006128(n).
The sum of row n is A066186(n).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
First differs from both A026791 and A080576 at a(17) = T(4,7).
Cf. A000041, A027750, A176206, A221529, A336812, A338156.

nrows=10; Array[Sort[Flatten[IntegerPartitions[#]]]&,nrows] (* Paolo Xausa, Jun 16 2022 *)

row(n) = my(list = List()); forpart(p=n, for (k=1, #p, listput(list, p[k]));); vecsort(Vec(list)); \\ Michel Marcus, Jun 16 2022

cp's OEIS Frontend

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