This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340057 #35 Mar 15 2024 21:27:28
%S A340057 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 A340057 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 A340057 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 A340057 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 A340057 This triangle is a condensed version of the more irregular triangle A340035.
%C A340057 For further information about the correspondence divisor/part see A338156.
%H A340057 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 A340057 Triangle begins:
%e A340057 [1];
%e A340057 [1], [1, 2];
%e A340057 [2], [1, 2], [1, 3];
%e A340057 [3], [2, 4], [1, 3], [1, 2, 4];
%e A340057 [5], [3, 6], [2, 6], [1, 2, 4], [1, 5];
%e A340057 [7], [5, 10], [3, 9], [2, 4, 8], [1, 5], [1, 2, 3, 6];
%e A340057 [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
%e A340057 ...
%e A340057 Row sums gives A066186.
%e A340057 Written as a tetrahedrons the first five slices are:
%e A340057 --
%e A340057 1;
%e A340057 --
%e A340057 1,
%e A340057 1, 2;
%e A340057 -----
%e A340057 2,
%e A340057 1, 2,
%e A340057 1, 3;
%e A340057 -----
%e A340057 3,
%e A340057 2, 4,
%e A340057 1, 3,
%e A340057 1, 2, 4;
%e A340057 --------
%e A340057 5,
%e A340057 3, 6,
%e A340057 2, 6,
%e A340057 1, 2, 4,
%e A340057 1, 5;
%e A340057 --------
%e A340057 Row sums give A221529.
%e A340057 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 A340057 .
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 | n | | 1 | 2 | 3 | 4 | 5 |
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 | | - | | | | | 5 |
%e A340057 | C | - | | | | 3 | 3 6 |
%e A340057 | O | - | | | 2 | 2 4 | 2 6 |
%e A340057 | N | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e A340057 | D | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 .
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 | | A027750 | | | | | 1 |
%e A340057 | | A027750 | | | | | 1 |
%e A340057 | | A027750 | | | | | 1 |
%e A340057 | | A027750 | | | | | 1 |
%e A340057 | D | A027750 | | | | | 1 |
%e A340057 | I |---------|-----|-------|---------|-----------|-------------|
%e A340057 | V | A027750 | | | | 1 | 1 2 |
%e A340057 | I | A027750 | | | | 1 | 1 2 |
%e A340057 | S | A027750 | | | | 1 | 1 2 |
%e A340057 | O |---------|-----|-------|---------|-----------|-------------|
%e A340057 | R | A027750 | | | 1 | 1 2 | 1 3 |
%e A340057 | S | A027750 | | | 1 | 1 2 | 1 3 |
%e A340057 | |---------|-----|-------|---------|-----------|-------------|
%e A340057 | | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e A340057 | |---------|-----|-------|---------|-----------|-------------|
%e A340057 | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 .
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e A340057 | | | = | = = | = = = | = = = = | = = = = = |
%e A340057 | L | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e A340057 | I | | * | * * | * * * | * * * * | * * * * * |
%e A340057 | N | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e A340057 | K | | | | |\| | |\|\| | |\|\|\| | |\|\|\|\| |
%e A340057 | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 .
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 | P | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e A340057 | A | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e A340057 | R | | | | 3 | 3 1 | 3 1 1 |
%e A340057 | T | | | | | 2 2 | 2 2 1 |
%e A340057 | I | | | | | 4 | 4 1 |
%e A340057 | T | | | | | | 3 2 |
%e A340057 | I | | | | | | 5 |
%e A340057 | O | | | | | | |
%e A340057 | N | | | | | | |
%e A340057 | S | | | | | | |
%e A340057 |---|---------|-----|-------|---------|-----------|-------------|
%e A340057 .
%e A340057 The upper zone is a condensed version of the "divisors" zone.
%e A340057 The above table is the table of A340056 upside down.
%t A340057 A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* _Paolo Xausa_, Sep 02 2023 *)
%Y A340057 Nonzero terms of A221649.
%Y A340057 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 A340057 nonn,tabf
%O A340057 1,4
%A A340057 _Omar E. Pol_, Dec 27 2020