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%I #39 Aug 26 2021 16:41:36
%S 1,2,3,1,5,2,1,9,3,2,1,1,17,6,3,2,2,1,1,33,11,6,4,2,2,2,1,2,1,65,22,
%T 11,7,5,3,3,2,2,2,1,2,1,1,1,129,43,22,13,9,7,5,4,3,3,3,2,2,1,2,1,2,1,
%U 1,1,1,1,257,86,43,26,18,12,10,8,6,5,4,4,3,3,3,2,3
%N Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.
%C The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.
%C So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.
%C Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.
%C For the definition of "width" see A249351.
%C T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).
%C T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.
%e Triangle begins:
%e 1;
%e 2;
%e 3, 1;
%e 5, 2, 1;
%e 9, 3, 2, 1, 1;
%e 17, 6, 3, 2, 2, 1, 1;
%e 33, 11, 6, 4, 2, 2, 2, 1, 2, 1;
%e 65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
%e 129, 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 |_| Semilength = 1
%e 1
%e Row 2: _
%e _| |
%e |_ _|
%e 2 Semilength = 2
%e .
%e Row 3: _
%e | |
%e _| |
%e _ _| _|
%e |_ _ _|1 Semilength = 4
%e 3
%e .
%e Row 4: _
%e | |
%e | |
%e | |
%e _ _| |
%e _| _ _|
%e | _|
%e _ _ _ _| | 1 Semilength = 8
%e |_ _ _ _ _|2
%e 5
%e .
%e Row 5: _
%e | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e | |
%e _ _ _| |
%e | _ _ _|
%e _| |
%e _| _|
%e _ _| _| Semilength = 16
%e | _ _|1 1
%e | | 2
%e _ _ _ _ _ _ _ _| |3
%e |_ _ _ _ _ _ _ _ _|
%e 9
%e .
%e The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.
%Y Row sums give A000079.
%Y Column 1 gives A094373.
%Y For the characteristic shape of sigma(A000040(n)) see A346871.
%Y For the characteristic shape of sigma(A000217(n)) see A346873.
%Y For the visualization of Mersenne numbers A000225 see A346874.
%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, A000225, A237591, A237593, A245092, A249351, A262626.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Aug 06 2021
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