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.

%I A340031 #38 Sep 28 2023 04:57:37
%S A340031 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,
%T A340031 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,
%U A340031 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
%N 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.
%C A340031 Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.
%H A340031 Paolo Xausa, <a href="/A340031/b340031.txt">Table of n, a(n) for n = 1..11552</a> (rows 1..17 of the triangle, flattened)
%e A340031 Triangle begins:
%e A340031 [1];
%e A340031 [1,2],      [1];
%e A340031 [1,0,3],    [1,2],    [1],    [1];
%e A340031 [1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
%e A340031 [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];
%e A340031 [...
%e A340031 Written as an irregular tetrahedron the first five slices are:
%e A340031 [1],
%e A340031 -------
%e A340031 [1, 2],
%e A340031 [1],
%e A340031 ----------
%e A340031 [1, 0, 3],
%e A340031 [1, 2],
%e A340031 [1],
%e A340031 [1];
%e A340031 -------------
%e A340031 [1, 2, 0, 4],
%e A340031 [1, 0, 3],
%e A340031 [1, 2],
%e A340031 [1, 2],
%e A340031 [1],
%e A340031 [1],
%e A340031 [1];
%e A340031 ----------------
%e A340031 [1, 0, 0, 0, 5],
%e A340031 [1, 2, 0, 4],
%e A340031 [1, 0, 3],
%e A340031 [1, 0, 3],
%e A340031 [1, 2],
%e A340031 [1, 2],
%e A340031 [1, 2],
%e A340031 [1],
%e A340031 [1],
%e A340031 [1],
%e A340031 [1],
%e A340031 [1];
%e A340031 .
%e A340031 The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
%e A340031 .
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 | n |         |  1  |   2   |    3    |     4     |      5      |
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 | P |         |     |       |         |           |             |
%e A340031 | A |         |     |       |         |           |             |
%e A340031 | R |         |     |       |         |           |             |
%e A340031 | T |         |     |       |         |           |  5          |
%e A340031 | I |         |     |       |         |           |  3 2        |
%e A340031 | T |         |     |       |         |  4        |  4 1        |
%e A340031 | I |         |     |       |         |  2 2      |  2 2 1      |
%e A340031 | O |         |     |       |  3      |  3 1      |  3 1 1      |
%e A340031 | N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
%e A340031 | S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 .
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 |   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
%e A340031 | L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
%e A340031 | I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
%e A340031 | N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
%e A340031 | K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
%e A340031 |   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
%e A340031 |   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 .
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 |   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
%e A340031 |   |---------|-----|-------|---------|-----------|-------------|
%e A340031 |   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
%e A340031 |   |---------|-----|-------|---------|-----------|-------------|
%e A340031 | D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
%e A340031 | I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
%e A340031 | V |---------|-----|-------|---------|-----------|-------------|
%e A340031 | I | A127093 |     |       |         |  1        |  1 2        |
%e A340031 | S | A127093 |     |       |         |  1        |  1 2        |
%e A340031 | O | A127093 |     |       |         |  1        |  1 2        |
%e A340031 | R |---------|-----|-------|---------|-----------|-------------|
%e A340031 | S | A127093 |     |       |         |           |  1          |
%e A340031 |   | A127093 |     |       |         |           |  1          |
%e A340031 |   | A127093 |     |       |         |           |  1          |
%e A340031 |   | A127093 |     |       |         |           |  1          |
%e A340031 |   | A127093 |     |       |         |           |  1          |
%e A340031 |---|---------|-----|-------|---------|-----------|-------------|
%e A340031 .
%e A340031 The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
%e A340031 .
%t A340031 A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
%t A340031 A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
%t A340031 Array[A340031row,7] (* _Paolo Xausa_, Sep 28 2023 *)
%Y A340031 Row sums give A066186.
%Y A340031 Nonzero terms gives A338156.
%Y A340031 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.
%K A340031 nonn,tabf
%O A340031 1,3
%A A340031 _Omar E. Pol_, Dec 26 2020