A352425 - OEIS

A352425

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

%I #41 Feb 22 2024 20:11:54

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

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

%e Triangle begins:

%e [1];

%e [2];

%e [3],

%e [4];

%e [5];

%e [6], [3, 2, 1];

%e [7];

%e [8];

%e [9], [4, 3, 2];

%e [10];

%e [11];

%e [12], [5, 4, 3];

%e [13];

%e [14];

%e [15], [6, 5, 4], [5, 4, 3, 2, 1];

%e [16];

%e [17];

%e [18], [7, 6, 5];

%e [19];

%e [20], [6, 5, 4, 3, 2];

%e [21], [8, 7, 6];

%e [22];

%e [23];

%e [24], [9, 8, 7];

%e [25], [7, 6, 5, 4, 3];

%e [26];

%e [27], [10, 9, 8];

%e [28], [7, 6, 5, 4, 3, 2, 1];

%e ...

%e In the diagram below the m-th staircase walk starts at row A000384(m).

%e 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 _

%e _|1|

%e _|2 |

%e _|3 |

%e _|4 |

%e _|5 _|

%e _|6 |3|

%e _|7 |2|

%e _|8 _|1|

%e _|9 |4 |

%e _|10 |3 |

%e _|11 _|2 |

%e _|12 |5 |

%e _|13 |4 |

%e _|14 _|3 _|

%e _|15 |6 |5|

%e _|16 |5 |4|

%e _|17 _|4 |3|

%e _|18 |7 |2|

%e _|19 |6 _|1|

%e _|20 _|5 |6 |

%e _|21 |8 |5 |

%e _|22 |7 |4 |

%e _|23 _|6 |3 |

%e _|24 |9 _|2 |

%e _|25 |8 |7 |

%e _|26 _|7 |6 |

%e _|27 |10 |5 _|

%e |28 |9 |4 |7|

%e ...

%e The diagram is infinite.

%e For more information about the diagram see A286000.

%Y Subsequence of A299765.

%Y Row sums give A352257.

%Y Column 1 gives A000027.

%Y Records give A000027.

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

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

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Mar 15 2022