This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A296508 #120 May 19 2022 16:43:49 %S A296508 1,3,2,2,7,0,3,3,11,1,0,4,0,4,15,0,0,5,3,5,9,0,9,0,6,0,0,6,23,5,0,0,7, %T A296508 0,0,7,12,0,12,0,8,7,1,0,8,31,0,0,0,0,9,0,0,0,9,35,2,0,2,0,10,0,0,0, %U A296508 10,39,0,3,0,0,11,5,0,5,0,11,18,0,0,0,18,0,12,0,0,0,0,12,47,13,0,0,0,0,13,0,5,0,0,13 %N A296508 Irregular triangle read by rows: T(n,k) is the size of the subpart that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned subpart is already associated to a previous peak or if there is no subpart adjacent to the k-th peak, with n >= 1, k >= 1. %C A296508 Conjecture: row n is formed by the odd-indexed terms of the n-th row of triangle A280850 together with the even-indexed terms of the same row but listed in reverse order. Examples: the 15th row of A280850 is [8, 8, 7, 0, 1] so the 15th row of this triangle is [8, 7, 1, 0, 8]. The 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0] so the 75h row of this triangle is [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38]. %C A296508 For the definition of "subparts" see A279387. %C A296508 For more information about the mentioned Dyck paths see A237593. %C A296508 T(n,k) could be called the "charge" of the k-th peak of the largest Dyck path of the symmetric representation of sigma(n). %C A296508 The number of zeros in row n is A238005(n). - _Omar E. Pol_, Sep 11 2021 %H A296508 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a> %e A296508 Triangle begins (rows 1..28): %e A296508 1; %e A296508 3; %e A296508 2, 2; %e A296508 7, 0; %e A296508 3, 3; %e A296508 11, 1, 0; %e A296508 4, 0, 4; %e A296508 15, 0, 0; %e A296508 5, 3, 5; %e A296508 9, 0, 9, 0; %e A296508 6, 0, 0, 6; %e A296508 23, 5, 0, 0; %e A296508 7, 0, 0, 7; %e A296508 12, 0, 12, 0; %e A296508 8, 7, 1, 0, 8; %e A296508 31, 0, 0, 0, 0; %e A296508 9, 0, 0, 0, 9; %e A296508 35, 2, 0, 2, 0; %e A296508 10, 0, 0, 0, 10; %e A296508 39, 0, 3, 0, 0; %e A296508 11, 5, 0, 5, 0, 11; %e A296508 18, 0, 0, 0, 18, 0; %e A296508 12, 0, 0, 0, 0, 12; %e A296508 47, 13, 0, 0, 0, 0; %e A296508 13, 0, 5, 0, 0, 13; %e A296508 21, 0, 0, 0 21, 0; %e A296508 14, 6, 0, 6, 0, 14; %e A296508 55, 0, 0, 1, 0, 0, 0; %e A296508 ... %e A296508 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 in the third quadrant as shown below in Figure 1: %e A296508 . _ _ %e A296508 . | | | | %e A296508 . | | | | %e A296508 . | | | | %e A296508 . 8 | | | | %e A296508 . | | | | %e A296508 . | | | | %e A296508 . | | | | %e A296508 . |_|_ _ _ |_|_ _ _ %e A296508 . | |_ _ 8 | |_ _ %e A296508 . |_ | |_ _ | %e A296508 . |_ |_ 7 |_| |_ %e A296508 . 8 |_ _| 1 |_ _| %e A296508 . | 0 | %e A296508 . |_ _ _ _ _ _ _ _ |_ _ _ _ _ _ _ _ %e A296508 . |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _| %e A296508 . 8 8 %e A296508 . %e A296508 . Figure 1. The symmetric Figure 2. After the dissection %e A296508 . representation of sigma(15) of the symmetric representation %e A296508 . has three parts of size 8 of sigma(15) into layers of %e A296508 . because every part contains width 1 we can see four subparts, %e A296508 . 8 cells, so the 15th row of so the 15th row of this triangle is %e A296508 . triangle A237270 is [8, 8, 8]. [8, 7, 1, 0, 8]. See also below. %e A296508 . %e A296508 Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050: %e A296508 . %e A296508 . 12 _ _ _ _ _ _ _ _ %e A296508 . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7 %e A296508 . | | |_ _ _ _ _ _ _| %e A296508 . 0 _| | | %e A296508 . |_ _|9 _ _ _ _ _ _ |_ _ 0 %e A296508 . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_ 0 %e A296508 . 0 _ _ _| | 0 _| | |_ _ _ _ _| | %e A296508 . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7 %e A296508 . | | 0 _ _| | 11 _ _ _ _ |_ | | | %e A296508 . | | | _ _| 1 _| _ _ _|_ _ _ 3 |_|_ _ 5 | | %e A296508 . | | | | 0 _|_| | |_ _ _| | | | | %e A296508 . | | | | | _ _| |_ _ 3 | | | | %e A296508 . | | | | | | 3 _ _ | | | | | | %e A296508 . | | | | | | | _|_ 1 | | | | | | %e A296508 . _|_| _|_| _|_| _|_| |_| _|_| _|_| _|_| _ %e A296508 . | | | | | | | | | | | | | | | | %e A296508 . | | | | | | |_|_ _ _| | | | | | | | %e A296508 . | | | | | | 2 |_ _|_ _| _| | | | | | | %e A296508 . | | | | |_|_ 2 |_ _ _| 0 _ _| | | | | | %e A296508 . | | | | 4 |_ 7 _| _ _|0 | | | | %e A296508 . | | |_|_ _ 0 |_ _ _ _ | _| _ _ _| | | | %e A296508 . | | 6 |_ |_ _ _ _|_ _ _ _| | 0 _| _ _ _|0 | | %e A296508 . |_|_ _ _ 0 |_ 4 |_ _ _ _ _| _| _| | _ _ _| | %e A296508 . 8 | |_ _ 0 | 15| _| _| | _ _ _| %e A296508 . |_ _ | |_ _ _ _ _ _ | |_ _| 0 _| | 0 %e A296508 . 7 |_| |_ |_ _ _ _ _ _|_ _ _ _ _ _| | 5 _| _| %e A296508 . 1 |_ _| 6 |_ _ _ _ _ _ _| _ _| _| 0 %e A296508 . 0 | 23| _ _| 0 %e A296508 . |_ _ _ _ _ _ _ _ | | 0 %e A296508 . |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| | %e A296508 . 8 |_ _ _ _ _ _ _ _ _| %e A296508 . 31 %e A296508 . %e A296508 The diagram contains 30 subparts equaling A060831(16), the total number of partitions of all positive integers <= 16 into consecutive parts. %e A296508 For the construction of the spiral see A239660. %e A296508 From _Omar E. Pol_, Nov 26 2020: (Start) %e A296508 Also consider the infinite double-staircases diagram defined in A335616 (see the theorem). For n = 15 the diagram with first 15 levels looks like this: %e A296508 . %e A296508 Level "Double-staircases" diagram %e A296508 . _ %e A296508 1 _|1|_ %e A296508 2 _|1 _ 1|_ %e A296508 3 _|1 |1| 1|_ %e A296508 4 _|1 _| |_ 1|_ %e A296508 5 _|1 |1 _ 1| 1|_ %e A296508 6 _|1 _| |1| |_ 1|_ %e A296508 7 _|1 |1 | | 1| 1|_ %e A296508 8 _|1 _| _| |_ |_ 1|_ %e A296508 9 _|1 |1 |1 _ 1| 1| 1|_ %e A296508 10 _|1 _| | |1| | |_ 1|_ %e A296508 11 _|1 |1 _| | | |_ 1| 1|_ %e A296508 12 _|1 _| |1 | | 1| |_ 1|_ %e A296508 13 _|1 |1 | _| |_ | 1| 1|_ %e A296508 14 _|1 _| _| |1 _ 1| |_ |_ 1|_ %e A296508 15 |1 |1 |1 | |1| | 1| 1| 1| %e A296508 . %e A296508 Starting from A196020 and after the algorithm described n A280850 and the conjecture applied to the above diagram we have a new diagram as shown below: %e A296508 . %e A296508 Level "Ziggurat" diagram %e A296508 . _ %e A296508 6 |1| %e A296508 7 _ | | _ %e A296508 8 _|1| _| |_ |1|_ %e A296508 9 _|1 | |1 1| | 1|_ %e A296508 10 _|1 | | | | 1|_ %e A296508 11 _|1 | _| |_ | 1|_ %e A296508 12 _|1 | |1 1| | 1|_ %e A296508 13 _|1 | | | | 1|_ %e A296508 14 _|1 | _| _ |_ | 1|_ %e A296508 15 |1 | |1 |1| 1| | 1| %e A296508 . %e A296508 The 15th row %e A296508 of A249351: [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1] %e A296508 The 15th row %e A296508 of A237270: [ 8, 8, 8 ] %e A296508 The 15th row %e A296508 of this seq: [ 8, 7, 1, 0, 8 ] %e A296508 The 15th row %e A296508 of A280851: [ 8, 7, 1, 8 ] %e A296508 . %e A296508 (End) %Y A296508 Row sums give A000203. %Y A296508 Row n has length A003056(n). %Y A296508 Column k starts in row A000217(k). %Y A296508 Nonzero terms give A280851. %Y A296508 The number of nonzero terms in row n is A001227(n). %Y A296508 The triangle with n rows contain A060831(n) nonzero terms. %Y A296508 Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A238005, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850. %K A296508 nonn,tabf %O A296508 1,2 %A A296508 _Omar E. Pol_, Feb 10 2018