A340031 - OEIS

A340031

Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

%I #38 Sep 28 2023 04:57:37

%S 1,1,2,1,1,0,3,1,2,1,1,1,2,0,4,1,0,3,1,2,1,2,1,1,1,1,0,0,0,5,1,2,0,4,

%T 1,0,3,1,0,3,1,2,1,2,1,2,1,1,1,1,1,1,2,3,0,0,6,1,0,0,0,5,1,2,0,4,1,2,

%U 0,4,1,0,3,1,0,3,1,0,3,1,2,1,2,1,2,1,2,1,2,1,1,1,1,1,1,1

%N Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

%C Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

%H Paolo Xausa, <a href="/A340031/b340031.txt">Table of n, a(n) for n = 1..11552</a> (rows 1..17 of the triangle, flattened)

%e Triangle begins:

%e [1];

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

%e [1,0,3], [1,2], [1], [1];

%e [1,2,0,4], [1,0,3], [1,2], [1,2], [1], [1], [1];

%e [1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];

%e [...

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

%e [1],

%e -------

%e [1, 2],

%e [1],

%e ----------

%e [1, 0, 3],

%e [1, 2],

%e [1],

%e [1];

%e -------------

%e [1, 2, 0, 4],

%e [1, 0, 3],

%e [1, 2],

%e [1],

%e [1];

%e ----------------

%e [1, 0, 0, 0, 5],

%e [1, 2, 0, 4],

%e [1, 0, 3],

%e [1, 2],

%e [1],

%e [1];

%e .

%e The following table formed by three zones shows the correspondence between divisors 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 | | A127093 | 1 | 1 2 | 1 0 3 | 1 2 0 4 | 1 0 0 0 5 |

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

%e | | A127093 | | 1 | 1 2 | 1 0 3 | 1 2 0 4 |

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

%e | D | A127093 | | | 1 | 1 2 | 1 0 3 |

%e | I | A127093 | | | 1 | 1 2 | 1 0 3 |

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

%e | I | A127093 | | | | 1 | 1 2 |

%e | S | A127093 | | | | 1 | 1 2 |

%e | O | A127093 | | | | 1 | 1 2 |

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

%e | S | A127093 | | | | | 1 |

%e | | A127093 | | | | | 1 |

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

%e .

%e The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.

%e .

%t A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];

%t A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];

%t Array[A340031row,7] (* _Paolo Xausa_, Sep 28 2023 *)

%Y Row sums give A066186.

%Y Nonzero terms gives A338156.

%Y Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Dec 26 2020