A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

1, 2, 3, 4, 5, 6, 3, 2, 1, 7, 8, 9, 4, 3, 2, 10, 11, 12, 5, 4, 3, 13, 14, 15, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 18, 7, 6, 5, 19, 20, 6, 5, 4, 3, 2, 21, 8, 7, 6, 22, 23, 24, 9, 8, 7, 25, 7, 6, 5, 4, 3, 26, 27, 10, 9, 8, 28, 7, 6, 5, 4, 3, 2, 1, 29, 30, 11, 10, 9, 8, 7, 6, 5, 4

Offset: 1

Author

Omar E. Pol, Mar 15 2022

Keywords

• nonn
• tabf

Comments

Conjecture: the total number of parts in all partitions of n into an odd number of consecutive parts equals the sum of odd divisors of n that are <= A003056(n). In other words: row n has A341309(n) terms.

The first partition with 2*m - 1 parts appears in the row A000384(m), m >= 1.

Examples

			Triangle begins:
   [1];
   [2];
   [3],
   [4];
   [5];
   [6], [3, 2, 1];
   [7];
   [8];
   [9], [4, 3, 2];
  [10];
  [11];
  [12], [5, 4, 3];
  [13];
  [14];
  [15], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17];
  [18], [7, 6, 5];
  [19];
  [20], [6, 5, 4, 3, 2];
  [21], [8, 7, 6];
  [22];
  [23];
  [24], [9, 8, 7];
  [25], [7, 6, 5, 4, 3];
  [26];
  [27], [10, 9, 8];
  [28], [7, 6, 5, 4, 3, 2, 1];
  ...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
                                                           _
                                                         _|1|
                                                       _|2  |
                                                     _|3    |
                                                   _|4      |
                                                 _|5       _|
                                               _|6        |3|
                                             _|7          |2|
                                           _|8           _|1|
                                         _|9            |4  |
                                       _|10             |3  |
                                     _|11              _|2  |
                                   _|12               |5    |
                                 _|13                 |4    |
                               _|14                  _|3   _|
                             _|15                   |6    |5|
                           _|16                     |5    |4|
                         _|17                      _|4    |3|
                       _|18                       |7      |2|
                     _|19                         |6     _|1|
                   _|20                          _|5    |6  |
                 _|21                           |8      |5  |
               _|22                             |7      |4  |
             _|23                              _|6      |3  |
           _|24                               |9       _|2  |
         _|25                                 |8      |7    |
       _|26                                  _|7      |6    |
     _|27                                   |10       |5   _|
    |28                                     |9        |4  |7|
...
The diagram is infinite.
For more information about the diagram see A286000.
		

Crossrefs

Subsequence of A299765.

Row sums give A352257.

Column 1 gives A000027.

Records give A000027.

Row n contains A082647(n) of the mentioned partitions.

Cf. A000384, A003056, A067742, A204217, A237048, A237591, A237593, A240542, A245092, A285574, A285901, A286000, A286001, A320051, A320137, A320142, A341309, A351824.