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%I #49 Jun 13 2021 03:25:21
%S 1,3,4,7,6,11,1,8,15,13,18,12,23,5,14,24,23,1,31,18,35,4,20,39,3,32,
%T 36,24,47,13,31,42,40,55,1,30,59,13,32,63,48,54,45,3,71,20,38,60,56,
%U 79,11,42,83,13,44,84,73,5,72,48,95,29,57,93,72,98,54,107,13,72,111,9,80,90,60,119,37,12
%N Irregular triangle read by rows: T(n,k) is the sum of the subparts in the k-th layer of the symmetric representation of sigma(n), if such a layer exists.
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e Triangle begins (first 15 rows):
%e 1;
%e 3;
%e 4;
%e 7;
%e 6;
%e 11, 1;
%e 8;
%e 15;
%e 13;
%e 18;
%e 12;
%e 23, 5;
%e 14;
%e 24;
%e 23, 1;
%e ...
%e For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
%e . _ _
%e . | | | |
%e . | | | |
%e . | | | |
%e . | | | |
%e . | | | |
%e . _ _ _| | _ _ _| |
%e . 28 _| _ _| 23 _| _ _ _|
%e . _| | _| _| |
%e . | _| | _| _|
%e . | _ _| | |_ _|
%e . _ _ _ _ _ _| | _ _ _ _ _ _| | 5
%e . |_ _ _ _ _ _ _| |_ _ _ _ _ _ _|
%e .
%e . Figure 1. The symmetric Figure 2. After the dissection
%e . representation of sigma(12) of the symmetric representation
%e . has only one part which of sigma(12) into layers of
%e . contains 28 cells, so width 1 we can see two "subparts"
%e . A000203(12) = 28. that contain 23 and 5 cells
%e . respectively, so the 12th row of
%e . this triangle is [23, 5].
%e .
%e For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) is constructed as shown below in Figure 3:
%e . _ _
%e . | | | |
%e . | | | |
%e . | | | |
%e . | | | |
%e . 8 | | 8 | |
%e . | | | |
%e . | | | |
%e . _ _ _|_| _ _ _|_|
%e . 8 _ _| | 7 _ _| |
%e . | _| | _ _|
%e . _| _| _| |_|
%e . |_ _| |_ _| 1
%e . 8 | 8 |
%e . _ _ _ _ _ _ _ _| _ _ _ _ _ _ _ _|
%e . |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
%e .
%e . Figure 3. The symmetric Figure 4. After the dissection
%e . representation of sigma(15) of the symmetric representation
%e . has three parts of size 8, of sigma(15) into layers of
%e . whose sum is 8 + 8 + 8 = 24, width 1 we can see four "subparts".
%e . so A000203(15) = 24. The first layer has three subparts
%e . whose sum is 8 + 7 + 8 = 23. The
%e . second layer has only one subpart
%e . of size 1, so the 15th row of this
%e . triangle is [23, 1].
%e .
%Y For the definition of "subparts" see A279387.
%Y For the triangle of subparts see A279391.
%Y Row sums give A000203.
%Y Row n has length A250068(n).
%Y Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250070, A261699, A262626.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Dec 12 2016