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

%I A352425 #41 Feb 22 2024 20:11:54
%S A352425 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,
%T A352425 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,
%U A352425 3,26,27,10,9,8,28,7,6,5,4,3,2,1,29,30,11,10,9,8,7,6,5,4
%N A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.
%C A352425 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.
%C A352425 The first partition with 2*m - 1 parts appears in the row A000384(m), m >= 1.
%e A352425 Triangle begins:
%e A352425    [1];
%e A352425    [2];
%e A352425    [3],
%e A352425    [4];
%e A352425    [5];
%e A352425    [6], [3, 2, 1];
%e A352425    [7];
%e A352425    [8];
%e A352425    [9], [4, 3, 2];
%e A352425   [10];
%e A352425   [11];
%e A352425   [12], [5, 4, 3];
%e A352425   [13];
%e A352425   [14];
%e A352425   [15], [6, 5, 4], [5, 4, 3, 2, 1];
%e A352425   [16];
%e A352425   [17];
%e A352425   [18], [7, 6, 5];
%e A352425   [19];
%e A352425   [20], [6, 5, 4, 3, 2];
%e A352425   [21], [8, 7, 6];
%e A352425   [22];
%e A352425   [23];
%e A352425   [24], [9, 8, 7];
%e A352425   [25], [7, 6, 5, 4, 3];
%e A352425   [26];
%e A352425   [27], [10, 9, 8];
%e A352425   [28], [7, 6, 5, 4, 3, 2, 1];
%e A352425   ...
%e A352425 In the diagram below the m-th staircase walk starts at row A000384(m).
%e A352425 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.
%e A352425                                                            _
%e A352425                                                          _|1|
%e A352425                                                        _|2  |
%e A352425                                                      _|3    |
%e A352425                                                    _|4      |
%e A352425                                                  _|5       _|
%e A352425                                                _|6        |3|
%e A352425                                              _|7          |2|
%e A352425                                            _|8           _|1|
%e A352425                                          _|9            |4  |
%e A352425                                        _|10             |3  |
%e A352425                                      _|11              _|2  |
%e A352425                                    _|12               |5    |
%e A352425                                  _|13                 |4    |
%e A352425                                _|14                  _|3   _|
%e A352425                              _|15                   |6    |5|
%e A352425                            _|16                     |5    |4|
%e A352425                          _|17                      _|4    |3|
%e A352425                        _|18                       |7      |2|
%e A352425                      _|19                         |6     _|1|
%e A352425                    _|20                          _|5    |6  |
%e A352425                  _|21                           |8      |5  |
%e A352425                _|22                             |7      |4  |
%e A352425              _|23                              _|6      |3  |
%e A352425            _|24                               |9       _|2  |
%e A352425          _|25                                 |8      |7    |
%e A352425        _|26                                  _|7      |6    |
%e A352425      _|27                                   |10       |5   _|
%e A352425     |28                                     |9        |4  |7|
%e A352425 ...
%e A352425 The diagram is infinite.
%e A352425 For more information about the diagram see A286000.
%Y A352425 Subsequence of A299765.
%Y A352425 Row sums give A352257.
%Y A352425 Column 1 gives A000027.
%Y A352425 Records give A000027.
%Y A352425 Row n contains A082647(n) of the mentioned partitions.
%Y A352425 Cf. A000384, A003056, A067742, A204217, A237048, A237591, A237593, A240542, A245092, A285574, A285901, A286000, A286001, A320051, A320137, A320142, A341309, A351824.
%K A352425 nonn,tabf
%O A352425 1,2
%A A352425 _Omar E. Pol_, Mar 15 2022