A346872 - OEIS

%I #39 Aug 26 2021 16:41:36

%S 1,2,3,1,5,2,1,9,3,2,1,1,17,6,3,2,2,1,1,33,11,6,4,2,2,2,1,2,1,65,22,

%T 11,7,5,3,3,2,2,2,1,2,1,1,1,129,43,22,13,9,7,5,4,3,3,3,2,2,1,2,1,2,1,

%U 1,1,1,1,257,86,43,26,18,12,10,8,6,5,4,4,3,3,3,2,3

%N Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.

%C The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.

%C So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.

%C Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.

%C For the definition of "width" see A249351.

%C T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).

%C T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.

%e Triangle begins:

%e     1;

%e     2;

%e     3,  1;

%e     5,  2,  1;

%e     9,  3,  2,  1, 1;

%e    17,  6,  3,  2, 2, 1, 1;

%e    33, 11,  6,  4, 2, 2, 2, 1, 2, 1;

%e    65, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;

%e   129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;

%e ...

%e Illustration of initial terms:

%e .

%e Row 1:  _

%e        |_|                              Semilength = 1

%e         1

%e Row 2:    _

%e         _| |

%e        |_ _|

%e          2                              Semilength = 2

%e .

%e Row 3:        _

%e              | |

%e             _| |

%e         _ _|  _|

%e        |_ _ _|1                         Semilength = 4

%e           3

%e .

%e Row 4:                _

%e                      | |

%e                      | |

%e                      | |

%e                   _ _| |

%e                 _|  _ _|

%e                |  _|

%e         _ _ _ _| | 1                    Semilength = 8

%e        |_ _ _ _ _|2

%e             5

%e .

%e Row 5:                                _

%e                                      | |

%e                                      | |

%e                                      | |

%e                                      | |

%e                                      | |

%e                                      | |

%e                                      | |

%e                                 _ _ _| |

%e                                |  _ _ _|

%e                               _| |

%e                             _|  _|

%e                         _ _|  _|        Semilength = 16

%e                        |  _ _|1 1

%e                        | | 2

%e         _ _ _ _ _ _ _ _| |3

%e        |_ _ _ _ _ _ _ _ _|

%e                 9

%e .

%e The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.

%Y Row sums give A000079.

%Y Column 1 gives A094373.

%Y For the characteristic shape of sigma(A000040(n)) see A346871.

%Y For the characteristic shape of sigma(A000217(n)) see A346873.

%Y For the visualization of Mersenne numbers A000225 see A346874.

%Y For the characteristic shape of sigma(A000384(n)) see A346875.

%Y For the characteristic shape of sigma(A000396(n)) see A346876.

%Y For the characteristic shape of sigma(A008588(n)) see A224613.

%Y Cf. A000203, A000225, A237591, A237593, A245092, A249351, A262626.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Aug 06 2021