A302800 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.

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, 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, 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

Offset: 1

Author

Omar E. Pol, Apr 13 2018

Keywords

• nonn
• tabf

Comments

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

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

Examples

			Triangle begins:
    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, 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;
   99, 37, 20, 11,  4;
  109, 41, 22, 13,  5;
  120, 45, 24, 14,  7;
  131, 49, 27, 15,  8,  1;
...
Illustration for n = 10:
We draw the first 10 rows of the infinite diagram described in A237591 as shown below:
Row                           _
1                           _| |
2                         _|  _|
3                       _|   | |
4                     _|    _| |
5                   _|     |  _|
6                 _|      _| | |
7               _|       |   | |
8             _|        _|  _| |
9           _|         |   |  _|
10         |_ _ _ _ _ _|_ _|_|_|
Area             35     13  6 1
.
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].
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.
		

Crossrefs

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

Row sums give A000217, n >= 1.

Column 1 gives A024206 without its initial zero.

Column 2 gives the partial sums of the A261348.

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