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Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.
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%I #39 Sep 03 2023 11:15:35

%S 1,1,2,1,1,3,1,2,2,1,2,4,1,3,2,4,3,1,5,1,2,4,2,6,3,6,5,1,2,3,6,1,5,2,

%T 4,8,3,9,5,10,7,1,7,1,2,3,6,2,10,3,6,12,5,15,7,14,11,1,2,4,8,1,7,2,4,

%U 6,12,3,15,5,10,20,7,21,11,22,15,1,3,9,1,2,4,8,2,14,3,6,9,18,5

%N Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

%C This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

%H Paolo Xausa, <a href="/A340056/b340056.txt">Table of n, a(n) for n = 1..11528</a> (rows 1..75 of the triangle, flattened)

%e Triangle begins:

%e [1];

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

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

%e [1, 2, 4], [1, 3], [2, 4], [3];

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

%e [...

%e The row sums of triangle give A066186.

%e Written as an irregular tetrahedron the first five slices are:

%e 1;

%e -----

%e 1, 2,

%e 1;

%e -----

%e 1, 3,

%e 1, 2,

%e 2;

%e --------

%e 1, 2, 4,

%e 1, 3,

%e 2, 4,

%e 3;

%e --------

%e 1, 5,

%e 1, 2, 4,

%e 2, 6,

%e 3, 6,

%e 5;

%e --------

%e The row sums of tetrahedron give A339106.

%e The slices of the tetrahedron appear in the following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):

%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 | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| |

%e | I | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |

%e | N | | * | * * | * * * | * * * * | * * * * * |

%e | K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |

%e | | | = | = = | = = = | = = = = | = = = = = |

%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 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 |---|---------|-----|-------|---------|-----------|-------------|

%e | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |

%e | C | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |

%e | O | - | | | 2 | 2 4 | 2 6 |

%e | N | - | | | | 3 | 3 6 |

%e | D | - | | | | | 5 |

%e |---|---------|-----|-------|---------|-----------|-------------|

%e .

%e The lower zone is a condensed version of the "divisors" zone.

%t A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m],{m,0,n-1}]];Array[A340056row,10] (* _Paolo Xausa_, Sep 01 2023 *)

%Y Nonzero terms of A340011.

%Y Row sums give A066186.

%Y Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Dec 27 2020