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%I #42 Oct 03 2023 10:04:25
%S 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,
%T 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,
%U 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
%N 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.
%C For further information about the correspondence divisor/part see A338156.
%H Paolo Xausa, <a href="/A340032/b340032.txt">Table of n, a(n) for n = 1..11552</a> (rows 1..17 of the triangle, flattened)
%e Triangle begins:
%e 1;
%e 1, 1, 2;
%e 1, 1, 1, 2, 1, 0, 3;
%e 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
%e 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;
%e ...
%e Written as an irregular tetrahedron the first five slices are:
%e 1;
%e --
%e 1,
%e 1, 2;
%e -----
%e 1,
%e 1,
%e 1, 2,
%e 1, 0, 3;
%e --------
%e 1,
%e 1,
%e 1,
%e 1, 2,
%e 1, 2,
%e 1, 0, 3,
%e 1, 2, 0, 4;
%e -----------
%e 1,
%e 1,
%e 1,
%e 1,
%e 1,
%e 1, 2,
%e 1, 2,
%e 1, 2,
%e 1, 0, 3,
%e 1, 0, 3,
%e 1, 2, 0, 4,
%e 1, 0, 0, 0, 5;
%e --------------
%e ...
%e 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):
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | n | | 1 | 2 | 3 | 4 | 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | D | A127093 | | | | | 1 |
%e | I |---------|-----|-------|---------|-----------|-------------|
%e | V | A127093 | | | | 1 | 1 2 |
%e | I | A127093 | | | | 1 | 1 2 |
%e | S | A127093 | | | | 1 | 1 2 |
%e | O |---------|-----|-------|---------|-----------|-------------|
%e | R | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | S | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e | | | = | = = | = = = | = = = = | = = = = = |
%e | L | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e | I | | * | * * | * * * | * * * * | * * * * * |
%e | N | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e | K | | | | |\| | |\|\| | |\|\|\| | |\|\|\|\| |
%e | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | P | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e | A | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e | R | | | | 3 | 3 1 | 3 1 1 |
%e | T | | | | | 2 2 | 2 2 1 |
%e | I | | | | | 4 | 4 1 |
%e | T | | | | | | 3 2 |
%e | I | | | | | | 5 |
%e | O | | | | | | |
%e | N | | | | | | |
%e | S | | | | | | |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e The table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
%e Also the above table is the table of A340031 upside down.
%t A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
%t A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
%t Array[A340032row,7] (* _Paolo Xausa_, Sep 28 2023 *)
%Y Row sums give A066186.
%Y Nonzero terms gives A340035.
%Y 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.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Dec 26 2020
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