The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #51 Sep 28 2023 05:08:08
%S 1,1,2,1,1,0,3,1,2,2,1,2,0,4,1,0,3,2,4,3,1,0,0,0,5,1,2,0,4,2,0,6,3,6,
%T 5,1,2,3,0,0,6,1,0,0,0,5,2,4,0,8,3,0,9,5,10,7,1,0,0,0,0,0,7,1,2,3,0,0,
%U 6,2,0,0,0,10,3,6,0,12,5,0,15,7,14,11,1,2,0,4,0,0,0,8
%N Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.
%C This triangle is a condensed version of the more irregular triangle A340031.
%C For further information about the correspondence divisor/part see A338156.
%H Paolo Xausa, <a href="/A340011/b340011.txt">Table of n, a(n) for n = 1..11480</a> (rows 1..40 of the triangle, flattened)
%e Triangle begins:
%e [1];
%e [1, 2], [1];
%e [1, 0, 3], [1, 2], [2];
%e [1, 2, 0, 4], [1, 0, 3], [2, 4], [3];
%e [1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
%e [...
%e Row sums give A066186.
%e Written as an irregular tetrahedron the first five slices are:
%e --
%e 1;
%e -----
%e 1, 2,
%e 1;
%e --------
%e 1, 0, 3,
%e 1, 2,
%e 2;
%e -----------
%e 1, 2, 0, 4,
%e 1, 0, 3,
%e 2, 4,
%e 3;
%e --------------
%e 1, 0, 0, 0, 5,
%e 1, 2, 0, 4,
%e 2, 0, 6,
%e 3, 6,
%e 5;
%e --------------
%e Row sums give A339106.
%e The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | n | | 1 | 2 | 3 | 4 | 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | P | | | | | | |
%e | A | | | | | | |
%e | R | | | | | | |
%e | T | | | | | | 5 |
%e | I | | | | | | 3 2 |
%e | T | | | | | 4 | 4 1 |
%e | I | | | | | 2 2 | 2 2 1 |
%e | O | | | | 3 | 3 1 | 3 1 1 |
%e | N | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e | S | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| |
%e | I | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e | N | | * | * * | * * * | * * * * | * * * * * |
%e | K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e | | | = | = = | = = = | = = = = | = = = = = |
%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | D | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | I | A127093 | | | 1 | 1 2 | 1 0 3 |
%e | V |---------|-----|-------|---------|-----------|-------------|
%e | I | A127093 | | | | 1 | 1 2 |
%e | S | A127093 | | | | 1 | 1 2 |
%e | O | A127093 | | | | 1 | 1 2 |
%e | R |---------|-----|-------|---------|-----------|-------------|
%e | S | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e | | A127093 | | | | | 1 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |
%e | C | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |
%e | O | - | | | 2 | 2 4 | 2 0 6 |
%e | N | - | | | | 3 | 3 6 |
%e | D | - | | | | | 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e This lower zone of the table is a condensed version of the "divisors" zone.
%t A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
%t A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
%t Array[A340011row,10] (* _Paolo Xausa_, Sep 28 2023 *)
%Y Row sums give A066186.
%Y Nonzero terms give A340056.
%Y 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, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Dec 26 2020