OFFSET
1,2
COMMENTS
The characteristic shape of the symmetric representation of sigma(A000217(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, or vice versa, the smallest Dyck path has a peak and the largest Dyck path has valley.
So knowing this characteristic shape we can know if a number is a triangular number (or not) just by looking at the diagram, even ignoring the concept of triangular number.
Therefore we can see a geometric pattern of the distribution of the triangular 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(A000217(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000217(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th triangular number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th triangular number into exactly k + 1 consecutive parts.
FORMULA
EXAMPLE
Triangle begins:
1;
2, 1;
4, 1, 1;
6, 2, 1, 1;
8, 3, 2, 1, 1;
11, 4, 3, 1, 1, 1;
15, 5, 3, 2, 1, 1, 1;
19, 6, 4, 2, 2, 1, 1, 1;
23, 8, 5, 2, 2, 2, 1, 1, 1;
28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1;
40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
...
Illustration of initial terms:
Column T gives the triangular numbers (A000217).
Column S gives A074285, the sum of the divisors of the triangular numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
n T S Diagram
-------------------------------------------------------------------------
_ _ _ _ _ _ _
1 1 1 |_| | | | | | | | | | | | |
1 _ _|_| | | | | | | | | | |
2 3 4 |_ _| _ _| | | | | | | | | |
2 1| _| | | | | | | | |
_ _ _| _| _ _| | | | | | | |
3 6 12 |_ _ _ _| 1 | _ _| | | | | | |
4 1 _ _|_| | | | | | |
| _|1 _ _ _|_| | | | |
_ _ _ _ _| | 1 _ _| | | | | |
4 10 18 |_ _ _ _ _ _|2 | _| | | | |
6 _| _| _ _ _ _|_| | |
|_ _|1 1 | | | |
| 2 _| | | |
_ _ _ _ _ _ _ _|4 | _| _ _ _ _ _| |
3 15 24 |_ _ _ _ _ _ _ _| _ _|_| | _ _ _ _ _|
8 _ _| _|1 | |
|_ _ _|1 1 _ _| |
| 3 _ _| _ _|
|4 | _|
_ _ _ _ _ _ _ _ _ _ _| _| _|
4 21 32 |_ _ _ _ _ _ _ _ _ _ _| _ _ _| _|1 1
11 | _ _ _|2
| | 3
| |
| |5
_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
5 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
15
.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Aug 06 2021
EXTENSIONS
Name corrected by Omar E. Pol, Feb 06 2023
STATUS
approved