A299778 - OEIS

A299778

Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

%I #23 Jun 19 2019 17:57:24

%S 1,3,2,2,7,0,3,3,12,0,0,4,0,4,15,0,0,5,3,5,9,0,9,0,6,0,0,6,28,0,0,0,7,

%T 0,0,7,12,0,12,0,8,8,0,0,8,31,0,0,0,0,9,0,0,0,9,39,0,0,0,0,10,0,0,0,

%U 10,42,0,0,0,0,11,5,0,5,0,11,18,0,0,0,18,0,12,0,0,0,0,12,60,0,0,0,0,0,13,0,5,0,0,13

%N Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

%C For the definition of "part" of the symmetric representation of sigma see A237270.

%C For more information about the mentioned Dyck paths see A237593.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e Triangle begins (rows 1..28):

%e 1;

%e 3;

%e 2, 2;

%e 7, 0;

%e 3, 3;

%e 12, 0, 0;

%e 4, 0, 4;

%e 15, 0, 0;

%e 5, 3, 5;

%e 9, 0, 9, 0;

%e 6, 0, 0, 6;

%e 28, 0, 0, 0;

%e 7, 0, 0, 7;

%e 12, 0, 12, 0;

%e 8, 8, 0, 0, 8;

%e 31, 0, 0, 0, 0;

%e 9, 0, 0, 0, 9;

%e 39, 0, 0, 0, 0;

%e 10, 0, 0, 0, 10;

%e 42, 0, 0, 0, 0;

%e 11, 5, 0, 5, 0, 11;

%e 18, 0, 0, 0, 18, 0;

%e 12, 0, 0, 0, 0, 12;

%e 60, 0, 0, 0, 0, 0;

%e 13, 0, 5, 0, 0, 13;

%e 21, 0, 0, 0 21, 0;

%e 14, 6, 0, 6, 0, 14;

%e 56, 0, 0, 0, 0, 0, 0;

%e ...

%e Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:

%e .

%e . 12 _ _ _ _ _ _ _ _

%e . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7

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

%e . 0 _| | |

%e . |_ _|9 _ _ _ _ _ _ |_ _ 0

%e . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_ 0

%e . 0 _ _ _| | 0 _| | |_ _ _ _ _| |

%e . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7

%e . | | 0 _ _| | 12 _ _ _ _ |_ | | |

%e . | | | _ _| 0 _| _ _ _|_ _ _ 3 |_|_ _ 5 | |

%e . | | | | 0 _| | |_ _ _| | | | |

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

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

%e . | | | | | | | _|_ 1 | | | | | |

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

%e . | | | | | | | | | | | | | | | |

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

%e . | | | | | | 2 |_ _|_ _| _| | | | | | |

%e . | | | | |_|_ 2 |_ _ _| 0 _ _| | | | | |

%e . | | | | 4 |_ 7 _| _ _|0 | | | |

%e . | | |_|_ _ 0 |_ _ _ _ | _| _ _ _| | | |

%e . | | 6 |_ |_ _ _ _|_ _ _ _| | 0 _| _ _|0 | |

%e . |_|_ _ _ 0 |_ 4 |_ _ _ _ _| _| | _ _ _| |

%e . 8 | |_ _ 0 | 15| _| | _ _ _|

%e . |_ | |_ _ _ _ _ _ | _ _| 0 _| | 0

%e . 8 |_ |_ |_ _ _ _ _ _|_ _ _ _ _ _| | 0 _| _|

%e . 0 |_ _| 6 |_ _ _ _ _ _ _| _ _| _| 0

%e . 0 | 28| _ _| 0

%e . |_ _ _ _ _ _ _ _ | | 0

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

%e . 8 |_ _ _ _ _ _ _ _ _|

%e . 31

%e .

%e The diagram contains A237590(16) = 27 parts.

%e For the construction of the spiral see A239660.

%Y Row sums give A000203.

%Y Row n has length A003056(n).

%Y Column k starts in row A000217(k).

%Y Nonzero terms give A237270.

%Y The number of nonzero terms in row n is A237271(n).

%Y Column 1 is A241838.

%Y The triangle with n rows contain A237590(n) nonzero terms.

%Y Cf. A296508 (analog for subparts).

%Y Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850, A280851.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Apr 03 2018