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%I #139 Apr 01 2023 14:49:28
%S 1,1,2,1,1,3,1,2,1,1,1,2,4,1,3,1,2,1,2,1,1,1,1,5,1,2,4,1,3,1,3,1,2,1,
%T 2,1,2,1,1,1,1,1,1,2,3,6,1,5,1,2,4,1,2,4,1,3,1,3,1,3,1,2,1,2,1,2,1,2,
%U 1,2,1,1,1,1,1,1,1,1,7,1,2,3,6,1,5,1,5,1,2,4,1,2,4,1,2,4
%N Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.
%C In other words: in row n replace every term of n-th row of A176206 with its divisors.
%C The terms in row n are also all parts of all partitions of n.
%C As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
%C For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
%C From _Omar E. Pol_, Aug 01 2021: (Start)
%C The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.
%C The terms of row n in nonincreasing order give the n-th row of A302246.
%C The terms of row n in nondecreasing order give the n-th row of A302247.
%C For the connection with the tower described in A221529 see also A340035. (End)
%H Paolo Xausa, <a href="/A338156/b338156.txt">Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)</a>
%e Triangle begins:
%e [1];
%e [1,2], [1];
%e [1,3], [1,2], [1], [1];
%e [1,2,4], [1,3], [1,2], [1,2], [1], [1], [1];
%e [1,5], [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
%e ...
%e For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
%e Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
%e [1],
%e -------
%e [1, 2],
%e [1],
%e -------
%e [1, 3],
%e [1, 2],
%e [1],
%e [1];
%e ----------
%e [1, 2, 4],
%e [1, 3],
%e [1, 2],
%e [1, 2],
%e [1],
%e [1],
%e [1];
%e ----------
%e [1, 5],
%e [1, 2, 4],
%e [1, 3],
%e [1, 3],
%e [1, 2],
%e [1, 2],
%e [1, 2],
%e [1],
%e [1],
%e [1],
%e [1],
%e [1];
%e .
%e The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
%e The table is infinite. It is formed by three zones as follows:
%e The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
%e The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
%e Finally the middle zone shows the connection between the upper zone and the lower zone.
%e For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
%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 | | | | | |/| | |/|/| | |/ |/|/| | |/ | /|/|/| |
%e | L | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e | I | | * | * * | * * * | * * * * | * * * * * |
%e | N | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e | K | | = | = = | = = = | = = = = | = = = = = |
%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e | | | | | |\| | |\|\| | |\ |\|\| | |\ |\ |\|\| |
%e | | A206561 | 1 | 4 2 | 9 5 3 | 20 13 7 4 | 35 23 15 9 5 |
%e |---|---------|-----|-------|---------|------------|---------------|
%e .
%e |---|---------|-----|-------|---------|------------|---------------|
%e | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
%e | |---------|-----|-------|---------|------------|---------------|
%e | | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e | |---------|-----|-------|---------|------------|---------------|
%e | D | A027750 | | | 1 | 1 2 | 1 3 |
%e | I | A027750 | | | 1 | 1 2 | 1 3 |
%e | V |---------|-----|-------|---------|------------|---------------|
%e | I | A027750 | | | | 1 | 1 2 |
%e | S | A027750 | | | | 1 | 1 2 |
%e | O | A027750 | | | | 1 | 1 2 |
%e | R |---------|-----|-------|---------|------------|---------------|
%e | S | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e | | A027750 | | | | | 1 |
%e |---|---------|-----|-------|---------|------------|---------------|
%e .
%e Note that every row in the lower zone lists A027750.
%e Also the lower zone for every positive integer can be constructued using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5] then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
%e Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
%e The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
%e The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.
%t A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
%t A338156[10] (* Generates 10 rows *) (* _Paolo Xausa_, Jan 12 2023 *)
%o (PARI)
%o A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
%o A338156(10) \\ Generates 10 rows - _Paolo Xausa_, Feb 17 2023
%Y Nonzero terms of A340031.
%Y Row n has length A006128(n).
%Y The sum of row n is A066186(n).
%Y The product of row n is A007870(n).
%Y Row n lists the first n rows of A336812 (a subsequence).
%Y The number of parts k in row n is A066633(n,k).
%Y The sum of all parts k in row n is A138785(n,k).
%Y The number of parts >= k in row n is A181187(n,k).
%Y The sum of all parts >= k in row n is A206561(n,k).
%Y The number of parts <= k in row n is A210947(n,k).
%Y The sum of all parts <= k in row n is A210948(n,k).
%Y Cf. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Oct 14 2020