%I #35 Mar 15 2024 21:27:28
%S 1,1,1,2,2,1,2,1,3,3,2,4,1,3,1,2,4,5,3,6,2,6,1,2,4,1,5,7,5,10,3,9,2,4,
%T 8,1,5,1,2,3,6,11,7,14,5,15,3,6,12,2,10,1,2,3,6,1,7,15,11,22,7,21,5,
%U 10,20,3,15,2,4,6,12,1,7,1,2,4,8,22,15,30,11,33,7,14,28,5,25
%N Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.
%C This triangle is a condensed version of the more irregular triangle A340035.
%C For further information about the correspondence divisor/part see A338156.
%H Paolo Xausa, <a href="/A340057/b340057.txt">Table of n, a(n) for n = 1..11528</a> (rows 1..75 of the triangle, flattened)
%e Triangle begins:
%e [1];
%e [1], [1, 2];
%e [2], [1, 2], [1, 3];
%e [3], [2, 4], [1, 3], [1, 2, 4];
%e [5], [3, 6], [2, 6], [1, 2, 4], [1, 5];
%e [7], [5, 10], [3, 9], [2, 4, 8], [1, 5], [1, 2, 3, 6];
%e [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
%e ...
%e Row sums gives A066186.
%e Written as a tetrahedrons the first five slices are:
%e --
%e 1;
%e --
%e 1,
%e 1, 2;
%e -----
%e 2,
%e 1, 2,
%e 1, 3;
%e -----
%e 3,
%e 2, 4,
%e 1, 3,
%e 1, 2, 4;
%e --------
%e 5,
%e 3, 6,
%e 2, 6,
%e 1, 2, 4,
%e 1, 5;
%e --------
%e Row sums give A221529.
%e 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):
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | n | | 1 | 2 | 3 | 4 | 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | - | | | | | 5 |
%e | C | - | | | | 3 | 3 6 |
%e | O | - | | | 2 | 2 4 | 2 6 |
%e | N | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e | D | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
%e |---|---------|-----|-------|---------|-----------|-------------|
%e .
%e |---|---------|-----|-------|---------|-----------|-------------|
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | D | A027750 | | | | | 1 |
%e | I |---------|-----|-------|---------|-----------|-------------|
%e | V | A027750 | | | | 1 | 1 2 |
%e | I | A027750 | | | | 1 | 1 2 |
%e | S | A027750 | | | | 1 | 1 2 |
%e | O |---------|-----|-------|---------|-----------|-------------|
%e | R | A027750 | | | 1 | 1 2 | 1 3 |
%e | S | A027750 | | | 1 | 1 2 | 1 3 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e | |---------|-----|-------|---------|-----------|-------------|
%e | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 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 upper zone is a condensed version of the "divisors" zone.
%e The above table is the table of A340056 upside down.
%t A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* _Paolo Xausa_, Sep 02 2023 *)
%Y Nonzero terms of A221649.
%Y Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Dec 27 2020