|
|
A221649
|
|
Tetrahedron E(n,j,k) = k*T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.
|
|
12
|
|
|
1, 1, 1, 2, 2, 1, 2, 1, 0, 3, 3, 2, 4, 1, 0, 3, 1, 2, 0, 4, 5, 3, 6, 2, 0, 6, 1, 2, 0, 4, 1, 0, 0, 0, 5, 7, 5, 10, 3, 0, 9, 2, 4, 0, 8, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 11, 7, 14, 5, 0, 15, 3, 6, 0, 12, 2, 0, 0, 0, 10, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
The tetrahedron shows a connection between divisors and partitions.
The sum of all elements of slice n is A066186(n).
The sum of row j of slice n is A221529(n,j).
The sum of column k of slice n is A138785(n,k), the sum of all parts of size k in all partitions of n.
See also the tetrahedron of A221650.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
First five slices of tetrahedron are
---------------------------------------------------
---------------------------------------------------
1 1 1, 1 1
...................................................
2 1 1, 1
2 2 1, 2, 3 4
...................................................
3 1 2, 2
3 2 1, 2, 3
3 3 1, 0, 3, 4 9
...................................................
4 1 3, 3
4 2 2, 4, 6
4 3 1, 0, 3, 4
4 4 1, 2, 0, 4, 7 20
...................................................
5 1 5, 5
5 2 3, 6, 9
5, 3, 2, 0, 6, 8
5, 4, 1, 2, 0, 4, 7
5, 5, 1, 0, 0, 0, 5, 6 35
...................................................
.
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 0 6 |
| N | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
| D | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| I |---------|-----|-------|---------|-----------|-------------|
| 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 upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340011 upside down.
For more information about the correspondence divisor/part see A338156. (End)
|
|
MATHEMATICA
|
A221649row[n_]:=Flatten[Table[If[Divisible[j, k], PartitionsP[n-j]k, 0], {j, n}, {k, j}]]; Array[A221649row, 10] (* Paolo Xausa, Sep 26 2023 *)
|
|
CROSSREFS
|
Cf. A000005, A000041, A000203, A027750, A051731, A066186, A127093, A138785, A221529, A221650, A237593, A336811, A336812, A338156, A340011, A340031, A340032, A340035, A340056.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(18)-a(19) and a(28)-a(29) corrected by Paolo Xausa, Sep 26 2023
|
|
STATUS
|
approved
|
|
|
|