OFFSET
1,2
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.
EXAMPLE
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
.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 06 2021
STATUS
approved