A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1.

1, 2, 1, 4, 2, 1, 8, 3, 2, 1, 1, 16, 6, 3, 2, 2, 1, 1, 32, 11, 6, 4, 2, 2, 2, 1, 2, 1, 64, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 256, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3

Offset: 1

Author

Omar E. Pol, Aug 06 2021

Keywords

• nonn
• tabf

Comments

The Mersenne number A000225(n) does not has a characteristic shape of its symmetric representation of sigma(A000225(n)). On the other hand, we can find that number in two ways in the symmetric representation of the powers of 2 as follows: the Mersenne numbers are the semilength of the smallest Dyck path and also they equals the area (or the number of cells) of the region of the diagram (see examples).

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

T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000225(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000225(n).

T(n,k) is also the difference between the total number of partitions of all positive integers <= Mersenne number A000225(n) into k consecutive parts, and the total number of partitions of all positive integers <= Mersenne number A000225(n) into k + 1 consecutive parts.

Examples

			Triangle begins:
    1;
    2,  1;
    4,  2,  1;
    8,  3,  2,  1, 1;
   16,  6,  3,  2, 2, 1, 1;
   32, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   64, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:
       0_                               Semilength = 0    Area = 1
       |_|
Row 2:
          _
       1_| |                            Semilength = 1    Area = 3
       |_ _|
.
Row 3:        _
             | |
         1  _| |
       2_ _|  _|                        Semilength = 3    Area = 7
       |_ _ _|
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
              1 _|  _ _|
          4   2|  _|                    Semilength = 7    Area = 15
        _ _ _ _| |
       |_ _ _ _ _|
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                         1 1_|  _|
                      2 _ _|  _|        Semilength = 15   Area = 31
                       |  _ _|
               8      3| |
        _ _ _ _ _ _ _ _| |
       |_ _ _ _ _ _ _ _ _|
.
		

Crossrefs

Row sums give A000225, n >= 1.

Column 1 gives A000079.

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 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, A237591, A237593, A245092, A249351, A262626.