A302800 - OEIS

%I #34 Apr 16 2018 09:09:20

%S 1,3,5,1,8,2,11,4,15,5,1,19,7,2,24,9,3,29,11,5,35,13,6,1,41,16,7,2,48,

%T 18,9,3,55,21,11,4,63,24,12,6,71,27,14,7,1,80,30,16,8,2,89,34,18,9,3,

%U 99,37,20,11,4,109,41,22,13,5,120,45,24,14,7,131,49,27,15,8,1,143,53,29,17,9,2

%N Irregular triangle read by rows: T(n,k) is the area of the k-th region of the diagram with n rows described in A237591.

%C Column k lists the partial sums of the k-th column of triangle A237591.

%C We can see this sequence in the front view of the pyramid described in A245092.

%e Triangle begins:

%e     1;

%e     3;

%e     5,  1;

%e     8,  2;

%e    11,  4;

%e    15,  5,  1;

%e    19,  7,  2;

%e    24,  9,  3;

%e    29, 11,  5;

%e    35, 13,  6,  1;

%e    41, 16,  7,  2;

%e    48, 18,  9,  3;

%e    55, 21, 11,  4;

%e    63, 24, 12,  6;

%e    71, 27, 14,  7,  1;

%e    80, 30, 16,  8,  2;

%e    89, 34, 18,  9,  3;

%e    99, 37, 20, 11,  4;

%e   109, 41, 22, 13,  5;

%e   120, 45, 24, 14,  7;

%e   131, 49, 27, 15,  8,  1;

%e ...

%e Illustration for n = 10:

%e We draw the first 10 rows of the infinite diagram described in A237591 as shown below:

%e Row                           _

%e 1                           _| |

%e 2                         _|  _|

%e 3                       _|   | |

%e 4                     _|    _| |

%e 5                   _|     |  _|

%e 6                 _|      _| | |

%e 7               _|       |   | |

%e 8             _|        _|  _| |

%e 9           _|         |   |  _|

%e 10         |_ _ _ _ _ _|_ _|_|_|

%e Area             35     13  6 1

%e .

%e The diagram contains four regions and the areas of the successives regions from left to right are respectively [35, 13, 6, 1], so the 10th row of this triangle is [35, 13, 6, 1].

%e Note that this infinite diagram gives a correspondence between the number of partitions into k consecutive parts and the symmetric representation of A000203, A024916, A004125 and many other integer sequences. For more information see A196020, A236104, A235791, A237048, A237593, A262626, A286000 and A286001.

%Y Row n has length A003056(n) hence column k starts in row A000217(k).

%Y Row sums give A000217, n >= 1.

%Y Column 1 gives A024206 without its initial zero.

%Y Column 2 gives the partial sums of the A261348.

%Y Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A286000, A286001.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Apr 13 2018