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Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.
5

%I #35 Mar 15 2024 21:27:28

%S 1,1,1,2,2,1,2,1,3,3,2,4,1,3,1,2,4,5,3,6,2,6,1,2,4,1,5,7,5,10,3,9,2,4,

%T 8,1,5,1,2,3,6,11,7,14,5,15,3,6,12,2,10,1,2,3,6,1,7,15,11,22,7,21,5,

%U 10,20,3,15,2,4,6,12,1,7,1,2,4,8,22,15,30,11,33,7,14,28,5,25

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

%C This triangle is a condensed version of the more irregular triangle A340035.

%C For further information about the correspondence divisor/part see A338156.

%H Paolo Xausa, <a href="/A340057/b340057.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], [1, 2];

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

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

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

%e [7], [5, 10], [3, 9], [2, 4, 8], [1, 5], [1, 2, 3, 6];

%e [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];

%e ...

%e Row sums gives A066186.

%e Written as a tetrahedrons the first five slices are:

%e --

%e 1;

%e --

%e 1,

%e 1, 2;

%e -----

%e 2,

%e 1, 2,

%e 1, 3;

%e -----

%e 3,

%e 2, 4,

%e 1, 3,

%e 1, 2, 4;

%e --------

%e 5,

%e 3, 6,

%e 2, 6,

%e 1, 2, 4,

%e 1, 5;

%e --------

%e Row sums give A221529.

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

%e .

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

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

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

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

%e | C | - | | | | 3 | 3 6 |

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

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

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

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

%e .

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

%e | | A027750 | | | | | 1 |

%e | | A027750 | | | | | 1 |

%e | | A027750 | | | | | 1 |

%e | | A027750 | | | | | 1 |

%e | D | A027750 | | | | | 1 |

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

%e | V | A027750 | | | | 1 | 1 2 |

%e | I | A027750 | | | | 1 | 1 2 |

%e | S | A027750 | | | | 1 | 1 2 |

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

%e | R | A027750 | | | 1 | 1 2 | 1 3 |

%e | S | A027750 | | | 1 | 1 2 | 1 3 |

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

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

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

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

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

%e .

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

%e | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |

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

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

%e | I | | * | * * | * * * | * * * * | * * * * * |

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

%e | K | | | | |\| | |\|\| | |\|\|\| | |\|\|\|\| |

%e | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |

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

%e .

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

%e | P | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |

%e | A | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |

%e | R | | | | 3 | 3 1 | 3 1 1 |

%e | T | | | | | 2 2 | 2 2 1 |

%e | I | | | | | 4 | 4 1 |

%e | T | | | | | | 3 2 |

%e | I | | | | | | 5 |

%e | O | | | | | | |

%e | N | | | | | | |

%e | S | | | | | | |

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

%e .

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

%e The above table is the table of A340056 upside down.

%t A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* _Paolo Xausa_, Sep 02 2023 *)

%Y Nonzero terms of A221649.

%Y Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Dec 27 2020