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%I A346874 #34 Aug 26 2021 15:18:01
%S A346874 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,
%T A346874 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,
%U A346874 1,1,256,86,43,26,18,12,10,8,6,5,4,4,3,3,3,2,3
%N A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1.
%C A346874 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).
%C A346874 Therefore we can see a geometric pattern of the distribution of the Mersenne numbers in the stepped pyramid described in A245092.
%C A346874 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).
%C A346874 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.
%e A346874 Triangle begins:
%e A346874     1;
%e A346874     2,  1;
%e A346874     4,  2,  1;
%e A346874     8,  3,  2,  1, 1;
%e A346874    16,  6,  3,  2, 2, 1, 1;
%e A346874    32, 11,  6,  4, 2, 2, 2, 1, 2, 1;
%e A346874    64, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
%e A346874   128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
%e A346874 ...
%e A346874 Illustration of initial terms:
%e A346874 .
%e A346874 Row 1:
%e A346874        0_                               Semilength = 0    Area = 1
%e A346874        |_|
%e A346874 Row 2:
%e A346874           _
%e A346874        1_| |                            Semilength = 1    Area = 3
%e A346874        |_ _|
%e A346874 .
%e A346874 Row 3:        _
%e A346874              | |
%e A346874          1  _| |
%e A346874        2_ _|  _|                        Semilength = 3    Area = 7
%e A346874        |_ _ _|
%e A346874 .
%e A346874 Row 4:                _
%e A346874                      | |
%e A346874                      | |
%e A346874                      | |
%e A346874                   _ _| |
%e A346874               1 _|  _ _|
%e A346874           4   2|  _|                    Semilength = 7    Area = 15
%e A346874         _ _ _ _| |
%e A346874        |_ _ _ _ _|
%e A346874 .
%e A346874 Row 5:                                _
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                      | |
%e A346874                                 _ _ _| |
%e A346874                                |  _ _ _|
%e A346874                               _| |
%e A346874                          1 1_|  _|
%e A346874                       2 _ _|  _|        Semilength = 15   Area = 31
%e A346874                        |  _ _|
%e A346874                8      3| |
%e A346874         _ _ _ _ _ _ _ _| |
%e A346874        |_ _ _ _ _ _ _ _ _|
%e A346874 .
%Y A346874 Row sums give A000225, n >= 1.
%Y A346874 Column 1 gives A000079.
%Y A346874 For the characteristic shape of sigma(A000040(n)) see A346871.
%Y A346874 For the characteristic shape of sigma(A000079(n)) see A346872.
%Y A346874 For the characteristic shape of sigma(A000217(n)) see A346873.
%Y A346874 For the characteristic shape of sigma(A000384(n)) see A346875.
%Y A346874 For the characteristic shape of sigma(A000396(n)) see A346876.
%Y A346874 For the characteristic shape of sigma(A008588(n)) see A224613.
%Y A346874 Cf. A000203, A237591, A237593, A245092, A249351, A262626.
%K A346874 nonn,tabf
%O A346874 1,2
%A A346874 _Omar E. Pol_, Aug 06 2021

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