OFFSET
1,1
COMMENTS
The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second 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(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts.
EXAMPLE
Triangle begins:
2, 1;
6, 2, 1, 1;
11, 4, 3, 1, 1, 1;
19, 6, 4, 2, 2, 1, 1, 1;
28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column h gives the n-th second hexagonal number (A014105).
Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
----------------------------------------------------------------------------
n h S Diagram
----------------------------------------------------------------------------
_ _ _
| | | | | |
_ _|_| | | | |
1 3 4 |_ _|1 | | | |
2 | | | |
_ _| | | |
| _ _| | |
_ _|_| | |
| _|1 | |
_ _ _ _ _| | 1 | |
2 10 18 |_ _ _ _ _ _|2 | |
6 _ _ _ _|_|
| |
_| |
| _|
_ _|_|
_ _| _|1
|_ _ _|1 1
| 3
|4
_ _ _ _ _ _ _ _ _ _ _| \
3 21 32 |_ _ _ _ _ _ _ _ _ _ _| \
11 |\
_| \
| \
_ _| _\
_ _| _| \
| _|1 \
_ _ _| _ _|1 1
| | 2
| _ _ _ _|2
| | 4
| |
| |6
| |
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
19
.
The symmetric representation of sigma(36) is partially illustrated because it is too big to include totally here.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 17 2021
STATUS
approved