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