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A340031
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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.
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8
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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
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OFFSET
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1,3
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COMMENTS
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Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.
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LINKS
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EXAMPLE
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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];
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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 |
|---|---------|-----|-------|---------|-----------|-------------|
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|---|---------|-----|-------|---------|-----------|-------------|
| | 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 |---------|-----|-------|---------|-----------|-------------|
| R |---------|-----|-------|---------|-----------|-------------|
|---|---------|-----|-------|---------|-----------|-------------|
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The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
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MATHEMATICA
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A127093row[n_]:=Table[Boole[Divisible[n, k]]k, {k, n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1], PartitionsP[m-1]], {m, n}]];
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CROSSREFS
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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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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