OFFSET
1,3
COMMENTS
This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)
EXAMPLE
Triangle begins:
[1];
[1, 2], [1];
[1, 3], [1, 2], [2];
[1, 2, 4], [1, 3], [2, 4], [3];
[1, 5], [1, 2, 4], [2, 6], [3, 6], [5];
[...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
1;
-----
1, 2,
1;
-----
1, 3,
1, 2,
2;
--------
1, 2, 4,
1, 3,
2, 4,
3;
--------
1, 5,
1, 2, 4,
2, 6,
3, 6,
5;
--------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in 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 |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
| |---------|-----|-------|---------|-----------|-------------|
| | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| |---------|-----|-------|---------|-----------|-------------|
| D | A027750 | | | 1 | 1 2 | 1 3 |
| I | A027750 | | | 1 | 1 2 | 1 3 |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A027750 | | | | 1 | 1 2 |
| S | A027750 | | | | 1 | 1 2 |
| O | A027750 | | | | 1 | 1 2 |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
| C | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| O | - | | | 2 | 2 4 | 2 6 |
| N | - | | | | 3 | 3 6 |
| D | - | | | | | 5 |
|---|---------|-----|-------|---------|-----------|-------------|
.
The lower zone is a condensed version of the "divisors" zone.
MATHEMATICA
A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m], {m, 0, n-1}]]; Array[A340056row, 10] (* Paolo Xausa, Sep 01 2023 *)
CROSSREFS
Nonzero terms of A340011.
Row sums give A066186.
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, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 27 2020
STATUS
approved