A346875 Irregular triangle read by rows in which row n lists the row A000384(n) of A237591, n >= 1.

1, 4, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 1, 23, 8, 5, 2, 2, 2, 1, 1, 1, 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1, 46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1, 61, 20, 11, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 77, 26, 14, 8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1, 96, 32, 16

Offset: 1

Author

Omar E. Pol, Aug 06 2021

Keywords

• nonn
• tabf

Comments

The characteristic shape of the symmetric representation of sigma(A000384(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.

So knowing this we can know if a number is a hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.

Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.

T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000384(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000384(n-1).

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

Examples

			Triangle begins:
   1;
   4,  1,  1;
   8,  3,  2,  1, 1;
  15,  5,  3,  2, 1, 1, 1;
  23,  8,  5,  2, 2, 2, 1, 1, 1;
  34, 11,  6,  4, 3, 2, 2, 1, 1, 1, 1;
  46, 16,  8,  5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
  61, 20, 11,  6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  77, 26, 14,  8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1;
  96, 32, 16, 10, 7, 5, 4, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column H gives the nonzero hexagonal numbers (A000384).
Column S gives the sum of the divisors of the hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    H    S   Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                 1        | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _| 1                 | |                       | |
                    4    1                  | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|1 1                              | |
                                | 2                                   | |
                 _ _ _ _ _ _ _ _|4                           _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                        8                                   | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|1 1
                                             _ _ _| | 1
                                            |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.
		

Crossrefs

Row sums give A000384, n >= 1.

Row lengths give A005408.

Column 1 is A267682, n >= 1.

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

For the characteristic shape of sigma(A000079(n)) see A346872.

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

For the visualization of Mersenne numbers A000225 see A346874.

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

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

Cf. A000203, A237591, A237593, A245092, A249351, A262626.