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%I #43 Jan 06 2024 14:31:59
%S 2,1,6,2,1,1,11,4,3,1,1,1,19,6,4,2,2,1,1,1,28,10,5,3,3,2,1,1,1,1,40,
%T 13,7,5,3,2,2,2,1,1,1,1,53,18,10,5,4,3,3,2,1,2,1,1,1,1,69,23,12,7,5,4,
%U 3,2,2,2,2,1,1,1,1,1,86,29,15,9,6,5,4,2,3,2,2,1,2,1,1,1,1,1
%N Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.
%C The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
%C So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
%C Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
%C T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).
%C T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts.
%e Triangle begins:
%e 2, 1;
%e 6, 2, 1, 1;
%e 11, 4, 3, 1, 1, 1;
%e 19, 6, 4, 2, 2, 1, 1, 1;
%e 28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
%e 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
%e 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
%e 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
%e 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
%e ...
%e Illustration of initial terms:
%e Column h gives the n-th second hexagonal number (A014105).
%e Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
%e ----------------------------------------------------------------------------
%e n h S Diagram
%e ----------------------------------------------------------------------------
%e _ _ _
%e | | | | | |
%e _ _|_| | | | |
%e 1 3 4 |_ _|1 | | | |
%e 2 | | | |
%e _ _| | | |
%e | _ _| | |
%e _ _|_| | |
%e | _|1 | |
%e _ _ _ _ _| | 1 | |
%e 2 10 18 |_ _ _ _ _ _|2 | |
%e 6 _ _ _ _|_|
%e | |
%e _| |
%e | _|
%e _ _|_|
%e _ _| _|1
%e |_ _ _|1 1
%e | 3
%e |4
%e _ _ _ _ _ _ _ _ _ _ _| \
%e 3 21 32 |_ _ _ _ _ _ _ _ _ _ _| \
%e 11 |\
%e _| \
%e | \
%e _ _| _\
%e _ _| _| \
%e | _|1 \
%e _ _ _| _ _|1 1
%e | | 2
%e | _ _ _ _|2
%e | | 4
%e | |
%e | |6
%e | |
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e 4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e 19
%e .
%e The symmetric representation of sigma(36) is partially illustrated because it is too big to include totally here.
%Y Row sums give A014105, n >= 1.
%Y Row lengths give A005843.
%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 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 For the characteristic shape of sigma(A174973(n)) see A317305.
%Y Cf. A000203, A237591, A237593, A245092, A249351, A262626.
%K nonn,tabf
%O 1,1
%A _Omar E. Pol_, Aug 17 2021