A341149 Irregular triangle read by rows T(n,k) in which row n lists n blocks where the m-th block consists of A000203(m) copies of A000041(n-m), with 1 <= m <= n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 5, 5, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 7, 7, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Author

Omar E. Pol, Feb 06 2021

Keywords

• nonn
• tabf

Comments

In the n-th row of the triangle the values of the m-th block are the number of cubes that are exactly below every cell of the symmetric representation of sigma(m) in the tower described in A221529 (see figure 5 in the example here).

Examples

			Triangle begins:
  1;
  1,1,1,1;
  2,1,1,1,1,1,1,1;
  3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
  5,3,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1;
  7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  ...
For n = 6 we have that:
                                 Row 6                    Row 6 of
m    A000203(m)  A000041(n-m)   block(m)                  A221529
1        1           7          [7]                           7
2        3           5          [5,5,5]                      15
3        4           3          [3,3,3,3]                    12
4        7           2          [2,2,2,2,2,2,2]              14
5        6           1          [1,1,1,1,1,1]                 6
6       12           1          [1,1,1,1,1,1,1,1,1,1,1,1]    12
.
so the 6th row of triangle is [7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] and the row sums equals A066186(6) = 66.
We can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1). For further information about these two associated objects see A221529.
       _ _ _ _ _ _
  11  |_ _ _      |              6
      |_ _ _|_    |        3     3
      |_ _    |   |          4   2
      |_ _|_ _|_  |      2   2   2      _
   7  |_ _ _    | |            5 1     | |
      |_ _ _|_  | |        3   2 1     |_|_
   5  |_ _    | | |          4 1 1     |   |
      |_ _|_  | | |      2   2 1 1     |_ _|_
   3  |_ _  | | | |        3 1 1 1     |_ _|_|_
   2  |_  | | | | |      2 1 1 1 1     |_ _ _|_|_ _
   1  |_|_|_|_|_|_|    1 1 1 1 1 1     |_ _ _ _|_|_|
.
        Figure 1.        Figure 2.       Figure 3.
       Front view       Partitions     Lateral view
      of the prism         of 6.       of the tower.
      of partitions.
.
                                                                      Row 6 of
                                        _ _ _ _ _ _                    A341148
                                    1  |_| | | |   |    7 5 3 2 1 1       19
                                    2  |_ _|_| |   |    5 5 3 2 1 1       17
                                    3  |_ _|  _|   |    3 3 2 2 1 1       12
                                    4  |_ _ _|    _|    2 2 2 1 1 1        9
                                    5  |        _|      1 1 1 1 1          5
                                    6  |_ _ _ _|        1 1 1 1            4
.
                                         Figure 4.       Figure 5.
                                         Top view         Heights
                                       of the tower.      in the
                                                         top view.
.
Figure 5 shows the heights of the terraces of the tower, or in other words the number of cubes in the column exactly below every cell of the top view. For example: in the 6th row of triangle the first block is [7] because there are seven cubes exactly below the symmetric representation of sigma(1) = 1. The second block is [5, 5, 5] because there are five cubes exactly below every cell of the symmetric representation of sigma(2) = 3. The third block is [3, 3, 3, 3] because there are three cubes exactly below every cell of the symmetric representation of sigma(3) = 4, and so on.
Note that the terraces that are the symmetric representation of sigma(5) and the terraces that are the symmetric representation of sigma(6) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12451 (rows 1..35 of triangle, flattened)

Crossrefs

Every column gives A000041.

Row lengths give A024916.

Row sums give the nonzero terms of A066186.

Cf. A000203, A221529, A236104, A237270, A237271, A237593, A337209, A339106, A340584, A341148, A345023.

Programs

Mathematica

A341149row[n_]:=Flatten[Array[ConstantArray[PartitionsP[n-#],DivisorSigma[1,#]]&,n]];
nrows=7;Array[A341149row,nrows] (* Paolo Xausa, Jun 20 2022 *)