A340035 -id:A340035 - cp's OEIS frontend

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

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

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| 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  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8

Offset: 1

Omar E. Pol, Dec 26 2020

This triangle is a condensed version of the more irregular triangle A340031.

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

			Triangle begins:
[1];
[1, 2],          [1];
[1, 0, 3],       [1, 2],       [2];
[1, 2, 0, 4],    [1, 0, 3],    [2, 4],    [3];
[1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
[...
Row sums give A066186.
Written as an irregular tetrahedron the first five slices are:
--
1;
-----
1, 2,
1;
--------
1, 0, 3,
1, 2,
2;
-----------
1, 2, 0, 4,
1, 0, 3,
2, 4,
3;
--------------
1, 0, 0, 0, 5,
1, 2, 0, 4,
2, 0, 6,
3, 6,
5;
--------------
Row sums give A339106.
The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| 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  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
| C | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
This lower zone of the table is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)

Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2020

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

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 0, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2,
  1, 0, 3;
  --------
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 2, 0, 4;
  -----------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 0, 3,
  1, 2, 0, 4,
  1, 0, 0, 0, 5;
  --------------
  ...
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 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 table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
Also the above table is the table of A340031 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A340035.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
Array[A340032row,7] (* Paolo Xausa, Sep 28 2023 *)

A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

Original entry on oeis.org

1, 4, 8, 1, 15, 4, 1, 21, 8, 4, 1, 1, 33, 15, 8, 4, 4, 1, 1, 41, 21, 15, 8, 8, 4, 4, 1, 1, 1, 1, 56, 33, 21, 15, 15, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 87, 56, 41, 33, 33, 21, 21, 15, 15, 15, 15, 8, 8, 8, 8

Offset: 1

Omar E. Pol, Jan 07 2021

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the tower described in A221529 whose largest side of the base is equal to n (see example). - Omar E. Pol, Jan 08 2022

			Triangle begins:
   1;
   4;
   8,  1;
  15,  4,  1;
  21,  8,  4,  1,  1;
  33, 15,  8,  4,  4,  1,  1;
  41, 21, 15,  8,  8,  4,  4, 1, 1, 1, 1;
  56, 33, 21, 15, 15,  8,  8, 4, 4, 4, 4, 1, 1, 1, 1;
  69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 9 the length of row 9 is A000041(9-1) = 22.
From _Omar E. Pol_, Jan 08 2022: (Start)
For n = 9 the lateral view and top view of the tower described in A221529 look like as shown below:
                        _
    22        1        | |
    21        1        | |
    20        1        | |
    19        1        | |
    18        1        | |
    17        1        | |
    16        1        |_|_
    15        4        |   |
    14        4        |   |
    13        4        |   |
    12        4        |_ _|_
    11        8        |   | |
    10        8        |   | |
     9        8        |   | |
     8        8        |_ _|_|_
     7       15        |     | |
     6       15        |_ _ _| |_
     5       21        |     |   |
     4       21        |_ _ _|_ _|_
     3       33        |_ _ _ _| | |_
     2       41        |_ _ _ _|_|_ _|_ _
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view
     k     T(9,k)        of the tower
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | |   |
                       |_ _|_| | | | |   |
                       |_ _|  _|_| | |   |
                       |_ _ _|    _|_|   |
                       |_ _ _|  _|    _ _|
                       |_ _ _ _|     |
                       |_ _ _ _|  _ _|
                       |         |
                       |_ _ _ _ _|
.
                           Top view
                         of the tower
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the tower, so T(9,1) = 69.
For n = 9 and k = 22 there is only one cubic cell in the level 22 (the top) of the tower, so T(9,22) = 1.
The volume of the tower (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000041 hence it's equal to A066186(9) = 270, equaling the sum of the 9th row of triangle. (End)

Row sums give A066186.
Row lengths give A000041.
The length of the m-th block in row n is A187219(m), m >= 1.
Cf. A350637 (analog for the stepped pyramid described in A245092).
Cf. A000203, A024916, A196020, A221529, A236104, A235791, A237270, A237271, A237593, A339278, A262626, A336811, A338156, A340035, A341149, A346533, A350333.

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; } \\ A336811
g(n) = sum(k=1, n, n\k*k); \\ A024916
row(n) = vector(f(n), k, g(T(n,k))); \\ Michel Marcus, Jan 22 2022

T(n,k) = A024916(A336811(n,k)).

T(n,k) = Sum_{j=1..n} A339278(j,k). - Omar E. Pol, Jan 08 2022

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Original entry on oeis.org

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1

Offset: 1

Omar E. Pol, Jan 12 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

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 *)

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38

Offset: 1

Omar E. Pol, Jul 22 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
Omar E. Pol, Illustration of the partitions of 10, prism and tower

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)

A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 2, 15, 8, 8, 3, 21, 15, 16, 12, 5, 33, 21, 30, 24, 20, 7, 41, 33, 42, 45, 40, 28, 11, 56, 41, 66, 63, 75, 56, 44, 15, 69, 56, 82, 99, 105, 105, 88, 60, 22, 87, 69, 112, 123, 165, 147, 165, 120, 88, 30, 99, 87, 138, 168, 205, 231, 231, 225, 176, 120, 42, 127, 99, 174

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture 1: T(n,k) is the sum of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
Conjecture 2: the sum of row n equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
Conjecture 3: T(n,k) is also the volume (or number of cubes) of the k-th block of a symmetric tower in which the terraces are the symmetric representation of sigma (n..1) starting from the base respectively (cf. A237270, A237593), hence the total area of the terraces is A024916(n), the same as the area of the base.
The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1). Hence the differences between levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
This symmetric tower has the property that its volume (or total number of cubes) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
For another symmetric tower of the same family and whose volume equals A066186(n) see A339106 and A221529.
The above three conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).

			Triangle begins:
   1;
   4,   1;
   8,   4,   2;
  15,   8,   8,   3;
  21,  15,  16,  12,   5;
  33,  21,  30,  24,  20,   7;
  41,  33,  42,  45,  40,  28,  11;
  56,  41,  66,  63,  75,  56,  44,  15;
  69,  56,  82,  99, 105, 105,  88,  60,  22;
  87,  69, 112, 123, 165, 147, 165, 120,  88,  30;
  99,  87, 138, 168, 205, 231, 231, 225, 176, 120,  42;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000041         T(6,k)
1      1  *  33   =   33
2      1  *  21   =   21
3      2  *  15   =   30
4      3  *   8   =   24
5      5  *   4   =   20
6      7  *   1   =    7
.          A024916
--------------------------
The sum of row 6 is 33 + 21 + 30 + 24 + 20 + 7 = 135, equaling A182738(6).

Columns 1 and 2 give A024916.
Column 3 gives A327329.
Leading diagonal gives A000041.
Row sums give A182738.
Cf. A000070, A066186, A176206, A221529, A221531, A237270, A237593, A336811, A336812, A338156, A339106, A340035,  A340424, A340425, A340426, A340524, A340526.

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.

A346530 a(n) is the number of faces of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

6, 6, 11, 14, 20, 27, 31, 38, 42, 51, 59

Offset: 1

Omar E. Pol, Jul 22 2021

The tower is a geometric object associated to all partitions of n.

The height of the tower equals A000041(n-1).

			For n = 1 the tower is a cube, and a cube has 6 faces, so a(1) = 6.

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346531 (number of edges), A346532 (number of vertices).
Cf. A325300 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

a(n) = A346531(n) - A346532(n) + 2 (Euler's formula).

cp's OEIS Frontend

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A340527 Triangle read by rows: T(n,k) = A024916(n-k+1)*A000041(k-1), 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author