A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

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

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.

Links

Table of n, a(n) for n=1..85.

Formula

T(n,k) = A237591(A000217(n),k). - Omar E. Pol, Feb 06 2023

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
.

Crossrefs

Row sums give A000217, n >= 1.

Column 1 gives A039823.

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

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

For the visualization of Mersenne numbers A000225 see A346874.

For the characteristic shape of sigma(A000384(n)) see A346875.

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

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

Cf. A000203, A074285, A237591, A237593, A245092, A249351, A262626.

Sequence in context: A118745 A235671 A131034 * A130313 A247073 A124428

Adjacent sequences: A346870 A346871 A346872 * A346874 A346875 A346876

Keyword

nonn,tabl

Author

Omar E. Pol, Aug 06 2021

Extensions

Name corrected by Omar E. Pol, Feb 06 2023

Status

approved