A181187 -id:A181187 - cp's OEIS frontend

A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

1, 2, 3, 3, 5, 6, 6, 9, 11, 12, 8, 14, 17, 19, 20, 15, 23, 29, 32, 34, 35, 19, 34, 42, 48, 51, 53, 54, 32, 51, 66, 74, 80, 83, 85, 86, 42, 74, 93, 108, 116, 122, 125, 127, 128, 64, 106, 138, 157, 172, 180, 186, 189, 191, 192, 83, 147, 189, 221, 240

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.

It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
--------------------------------------------------------
.  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  last       two        three      four       five
.  shell      shells     shells     shells     shells
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.    1        4+1        4+1        4+1        4+1
.      1      2+2+1      2+2+1      2+2+1      2+2+1
.      1        1+1      3+1+1      3+1+1      3+1+1
.        1        1+1      1+1+1    2+1+1+1    2+1+1+1
.          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
. ---------- ---------- ---------- ---------- ----------
.  8         14         17         19         20
.
So row 5 lists 8, 14, 17, 19, 20.
.
Triangle begins:
1;
2,    3;
3,    5,   6;
6,    9,  11,  12;
8,   14,  17,  19,  20;
15,  23,  29,  32,  34,  35;
19,  34,  42,  48,  51,  53,  54;
32,  51,  66,  74,  80,  83,  85,  86;
42,  74,  93, 108, 116, 122, 125, 127, 128;
64, 106, 138, 157, 172, 180, 186, 189, 191, 192;

Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

T(n,k) = A006128(n) - A006128(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138137(j).

A216053 a(n) is the position of the last two-tuple within the reverse lexicographic set of partitions of 2n and 2n+1, with a(1)-a(n) representing the positions of every 2-tuple partition of 2n and 2n+1.

Original entry on oeis.org

2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, 196, 273, 374, 509, 685, 916, 1213, 1598, 2088, 2715, 3507, 4509, 5764, 7339, 9297, 11733, 14743, 18461, 23026, 28630, 35472, 43821, 53964, 66274, 81157, 99134, 120771, 146786, 177971, 215309, 259892, 313066, 376327

Offset: 1

J. Stauduhar, Oct 12 2012

nonn

			With n = 3, 2n = 6.  The partitions of 6 are {{6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1,1}, {1,1,1,1,1,1}}.  The last 2-tuple is located at position 5. The positions of all 2-tuples are 2, 3, and 5.

J. Stauduhar, Table of n, a(n) for n = 1..10000

A diagonal of A181187.

Cf. A080577, A330661, A332706.

RecurrenceTable[{a[n+1] == a[n] + PartitionsP[(n)], a[1] == 2}, a, {n, 1, 44}]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, May 24 2018

a(n) = A330661(2n,2) = A330661(2n+1,2). - Alois P. Heinz, Feb 20 2020

A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 3, 2, 4, 1, 3, 1, 2, 4, 5, 3, 6, 2, 6, 1, 2, 4, 1, 5, 7, 5, 10, 3, 9, 2, 4, 8, 1, 5, 1, 2, 3, 6, 11, 7, 14, 5, 15, 3, 6, 12, 2, 10, 1, 2, 3, 6, 1, 7, 15, 11, 22, 7, 21, 5, 10, 20, 3, 15, 2, 4, 6, 12, 1, 7, 1, 2, 4, 8, 22, 15, 30, 11, 33, 7, 14, 28, 5, 25

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A340035.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  [1];
  [1],  [1, 2];
  [2],  [1, 2],  [1, 3];
  [3],  [2, 4],  [1, 3],  [1, 2, 4];
  [5],  [3, 6],  [2, 6],  [1, 2, 4],  [1, 5];
  [7],  [5, 10], [3, 9],  [2, 4, 8],  [1, 5],  [1, 2, 3, 6];
  [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
  ...
Row sums gives A066186.
Written as a tetrahedrons the first five slices are:
  --
  1;
  --
  1,
  1, 2;
  -----
  2,
  1, 2,
  1, 3;
  -----
  3,
  2, 4,
  1, 3,
  1, 2, 4;
  --------
  5,
  3, 6,
  2, 6,
  1, 2, 4,
  1, 5;
  --------
Row sums give A221529.
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   |    -    |     |       |         |           |  5          |
| C |    -    |     |       |         |  3        |  3 6        |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
| D | A027750 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 |     |       |         |  1        |  1 2        |
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 |     |       |  1      |  1 2      |  1   3      |
| S | A027750 |     |       |  1      |  1 2      |  1   3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340056 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A221649.

Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* Paolo Xausa, Sep 02 2023 *)

A207381 Total sum of the odd-indexed parts of all partitions of n.

Original entry on oeis.org

1, 3, 7, 14, 25, 45, 72, 117, 180, 275, 403, 596, 846, 1206, 1681, 2335, 3183, 4342, 5820, 7799, 10321, 13622, 17798, 23221, 30009, 38706, 49567, 63316, 80366, 101805, 128211, 161134, 201537, 251495, 312508, 387535, 478674, 590072, 724920, 888795, 1086324

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

For more information see A206563.

			For n = 5, write the partitions of 5 and below write the sums of their odd-indexed parts:
.    5
.    3+2
.    4+1
.    2+2+1
.    3+1+1
.    2+1+1+1
.    1+1+1+1+1
.  ------------
.   20 + 4 + 1 = 25
The total sum of the odd-indexed parts is 25 so a(5) = 25.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A206563, A207031, A207032, A207382.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[3]:
seq(a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0 , {1, 0, 0}, If[i < 1, {0, 0, 0},  g = b[n, i - 1]; h = If[i > n, {0, 0, 0}, b[n - i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]]; a[n_] := b[n, n][[3]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Dec 09 2016 after Alois P. Heinz *)

a(n) = A066186(n) - A207382(n) = A066897(n) + A207382(n).

More terms from Alois P. Heinz, Mar 12 2012

A207382 Sum of the even-indexed parts of all partitions of n.

Original entry on oeis.org

0, 1, 2, 6, 10, 21, 33, 59, 90, 145, 213, 328, 467, 684, 959, 1361, 1866, 2588, 3490, 4741, 6311, 8422, 11067, 14579, 18941, 24630, 31703, 40788, 52019, 66315, 83891, 106034, 133182, 167045, 208397, 259637, 321895, 398498, 491295, 604725, 741579, 908008

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

Also the sum of the floors of half the parts of all partitions of n, because the sum of one kind for a partition equals the sum of the other kind for the conjugate partition. Furthermore, this generalizes to taking m-th indices and dividing by m. - George Beck, Apr 15 2017

			For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:
. 5
. 3+2
. 4+1
. 2+2+1
. 3+1+1
. 2+1+1+1
. 1+1+1+1+1
------------
.   8 + 2   = 10
The sum of the even-indexed parts is 10, so a(5) = 10.
From _George Beck_, Apr 15 2017: (Start)
Alternatively, sum the floors of the parts divided by 2:
. 2
. 1+1
. 2+0
. 1+1+0
. 1+0+0
. 1+0+0+0
. 0+0+0+0+0
The sum is 10, so a(5) = 10. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

For more information see A206563.

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A207031, A207032, A207381.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[2]:
seq (a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0, 0}, i<1, {0, 0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];
Table [a[n], {n, 1, 50}] (* George Beck, Apr 15 2017 *)

a(n) = A066186(n) - A207381(n) = A207381(n) - A066897(n).

More terms from Alois P. Heinz, Mar 12 2012

A209918 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 26 2012

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each column of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the first row of each slice gives A058399.
It appears that the triangle formed by the last column of each slice gives A008284 and A058398.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

			---------------------------------------------------------
Illustration of first five                       A181187
slices of the tetrahedron                        Row sum
---------------------------------------------------------
. 1,                                                1
.    2, 1,                                          3
.       1,                                          1
.          3, 2, 1                                  6
.             1, 1,                                 2
.                1,                                 1
.                   5, 4, 2, 1,                    12
.                      1, 2, 2,                     5
.                         1, 1                      2
.                            1,                     1
.                               7, 6, 4, 2, 1,     20
.                                  1, 2, 3, 2,      8
.                                     1, 1, 2,      4
.                                        1, 1,      2
.                                           1,      1
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
.
Note that the 5th slice appears as one of three views of the model in the example section of A207380.

Row sums give A181187. Column sums give A209656. Main diagonal gives A210765. Another version is A209655.

Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A210961 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells mentioned in A210970.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 22 2012

			--------------------------------------------------------
Illustration of first five                      A210952
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    2, 1,                                         3
.          1,                                      1
.          2, 1,                                   3
.          3, 1, 1,                                5
.                   1,                             1
.                   2, 1,                          3
.                   5, 1, 1,                       7
.                   4, 3, 1, 1,                    9
.                               1,                 1
.                               2, 1,              3
.                               5, 1, 1,           7
.                               7, 3, 1, 1,       12
.                               5, 3, 2, 1, 1,    12
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Column sums give A181187.
Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
1;
1,2,1;
1,2,1,3,1,1;
1,2,1,5,1,1,4,3,1,1;
1,2,1,5,1,1,7,3,1,1,5,3,2,1,1;
1,2,1,5,1,1,9,3,1,1,12,5,2,1,1,6,6,4,2,1,1;
1,2,1,5,1,1,9,3,1,1,15,5,2,1,1,15,8,4,2,1,1,7,6,5,3,2,1,1;
In which row sums give A066186.

Cf. A066186, A135010, A138121, A181187, A210952, A210960, A210970.

A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 12, 11, 9, 6, 20, 19, 17, 14, 8, 35, 34, 32, 29, 23, 15, 54, 53, 51, 48, 42, 34, 19, 86, 85, 83, 80, 74, 66, 51, 32, 128, 127, 125, 122, 116, 108, 93, 74, 42, 192, 191, 189, 186, 180, 172, 157, 138, 106, 64, 275, 274, 272, 269, 263, 255, 240

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.

It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
--------------------------------------------------------
.  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  five       four       three      two        last
.  shells     shells     shells     shells     shell
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.  4+1        4+1        4+1        4+1          1
.  2+2+1      2+2+1      2+2+1      2+2+1          1
.  3+1+1      3+1+1      3+1+1        1+1          1
.  2+1+1+1    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
. ---------- ---------- ---------- ---------- ----------
. 20         19         17         14          8
.
So row 5 lists 20, 19, 17, 14, 8.
.
Triangle begins:
1;
3,     2;
6,     5,   3;
12,   11,   9,   6;
20,   19,  17,  14,  8;
35,   34,  32,  29,  23,  15;
54,   53,  51,  48,  42,  34,  19;
86,   85,  83,  80,  74,  66,  51,  32;
128, 127, 125, 122, 116, 108,  93,  74,  42;
192, 191, 189, 186, 180, 172, 157, 138, 106, 64;

Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

T(n,k) = A006128(n) - A006128(k-1).

T(n,k) = Sum_{j=k..n} A138137(j).

A330373 Sum of all parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 60, 80, 85, 90, 114, 140, 168, 176, 207, 264, 300, 312, 378, 448, 493, 540, 620, 736, 825, 884, 1015, 1188, 1295, 1406, 1599, 1840, 2009, 2184, 2451, 2772, 3060, 3312, 3666, 4176, 4557, 4900, 5457, 6084, 6625, 7182, 7920, 8792, 9576, 10324, 11328, 12540

Offset: 0

Omar E. Pol, Dec 17 2019

nonn

a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.
The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.
The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore, a(n) is also the sum of all parts of all partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

			For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
  * * * * *
  * *
  *
  *
  *
            * * * *
            * * *
            * *
            *
The sum of all parts of these partitions is 5 + 2 + 1 + 1 + 1 + 4 + 3 + 2 + 1 = 20, so a(10) = 20.
Also, in accordance with the first formula; a(10) = 2*10 = 20.

Freddy Barrera, Table of n, a(n) for n = 0..10000

Row sums of A330372.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.
Cf. A000700, A066186, A081362, A235324.

seq(n)={Vec(deriv(exp(sum(k=1, n, x^k/(k*(1 - (-x)^k)) + O(x*x^n)))), -(n+1))} \\ Andrew Howroyd, Dec 31 2019

a(n) = n*A000700(n).
a(n) = abs(n*A081362(n)).
a(n) = abs(A235324(n)), n >= 1.

A346741 Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 31 2021

The terms in row n are also all parts of all partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For further information about the correspondence divisor/part see A336811 and A338156.

			Triangle begins:
[1];
[1],[1, 2];
[1],[1, 2],[1, 3],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
...
Below the table shows the correspondence divisor/part.
|---|-----------------|-----|-------|---------|-----------|-------------|
| n |                 |  1  |   2   |    3    |     4     |      5      |
|---|-----------------|-----|-------|---------|-----------|-------------|
| P |                 |     |       |         |           |             |
| A |                 |     |       |         |           |             |
| R |                 |     |       |         |           |             |
| T |                 |     |       |         |           |  5          |
| I |                 |     |       |         |           |  3 2        |
| T |                 |     |       |         |  4        |  4 1        |
| I |                 |     |       |         |  2 2      |  2 2 1      |
| O |                 |     |       |  3      |  3 1      |  3 1 1      |
| N |                 |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |                 |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
----|-----------------|-----|-------|---------|-----------|-------------|
.
|---|-----------------|-----|-------|---------|-----------|-------------|
|   |         A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |                 |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I |         A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |                 |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K |         A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |                 |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   |         A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|-----------------|-----|-------|---------|-----------|-------------|
.
.   |-------|
.   |Section|
|---|-------|---------|-----|-------|---------|-----------|-------------|
|   |   1   | A000012 |  1  |  1    |  1      |  1        |  1          |
|   |-------|---------|-----|-------|---------|-----------|-------------|
|   |   2   | A000034 |     |  1 2  |  1 2    |  1 2      |  1 2        |
|   |-------|---------|-----|-------|---------|-----------|-------------|
| D |   3   | A010684 |     |       |  1   3  |  1   3    |  1   3      |
| I |       | A000012 |     |       |  1      |  1        |  1          |
| V |-------|---------|-----|-------|---------|-----------|-------------|
| I |   4   | A069705 |     |       |         |  1 2   4  |  1 2   4    |
| S |       | A000034 |     |       |         |  1 2      |  1 2        |
| O |       | A000012 |     |       |         |  1        |  1          |
| R |-------|---------|-----|-------|---------|-----------|-------------|
| S |   5   | A010686 |     |       |         |           |  1       5  |
|   |       | A010684 |     |       |         |           |  1   3      |
|   |       | A000034 |     |       |         |           |  1 2        |
|   |       | A000012 |     |       |         |           |  1          |
|   |       | A000012 |     |       |         |           |  1          |
|---|-------|---------|-----|-------|---------|-----------|-------------|
.
In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n.
The number of rows in the j-th section of the lower zone is equal to A000041(j-1).
The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.

Another version of A338156.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812.
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).
Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.

cp's OEIS Frontend

A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

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A216053 a(n) is the position of the last two-tuple within the reverse lexicographic set of partitions of 2n and 2n+1, with a(1)-a(n) representing the positions of every 2-tuple partition of 2n and 2n+1.

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A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

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A207381 Total sum of the odd-indexed parts of all partitions of n.

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A207382 Sum of the even-indexed parts of all partitions of n.

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A209918 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

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A210961 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells mentioned in A210970.

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A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

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A330373 Sum of all parts of all self-conjugate partitions of n.

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