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%I #29 Dec 07 2016 11:07:59
%S 3,12,9,9,12,12,39,18,18,21,21,72,27,27,30,30,96,36,36,39,15,39,120,
%T 45,45,48,48,144,54,36,54,57,57,84,84,63,63,66,66,234,72,72,75,21,75,
%U 108,108,81,81,84,48,84,120,120,90,90,93,93,312
%N Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-2).
%C Row n is a palindromic composition of sigma(4n-2).
%C Row n is also the row 4n-2 of A237270.
%C Row n has length A237271(4n-2).
%C Row sums give A239052.
%C Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the second quadrant of the spiral described in A239660, see example.
%C For the parts of the symmetric representation of sigma(4n-3), see A239931.
%C For the parts of the symmetric representation of sigma(4n-1), see A239933.
%C For the parts of the symmetric representation of sigma(4n), see A239934.
%C We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - _Omar E. Pol_, Dec 06 2016
%e The irregular triangle begins:
%e 3;
%e 12;
%e 9, 9;
%e 12, 12;
%e 39;
%e 18, 18;
%e 21, 21;
%e 72;
%e 27, 27;
%e 30, 30;
%e 96;
%e 36, 36;
%e 39, 15, 39;
%e 120;
%e 45, 45;
%e 48, 48;
%e ...
%e Illustration of initial terms in the second quadrant of the spiral described in A239660:
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e . | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e . | |
%e . | |
%e . | | _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e . _ _ _| | | _ _ _ _ _ _ _ _ _ _ _ _ _|
%e . | | | |
%e . _ _| _ _ _| | |
%e . 72 _| | | | _ _ _ _ _ _ _ _ _ _ _ _
%e . _| _| 21 _ _| | | _ _ _ _ _ _ _ _ _ _ _|
%e . | _| |_ _ _| | |
%e . _ _| _| _ _| | |
%e . | _ _| _| 18 _ _| | _ _ _ _ _ _ _ _ _ _
%e . | | | |_ _ _| | _ _ _ _ _ _ _ _ _|
%e . _ _ _ _ _| | 21 _ _| _| | |
%e . | _ _ _ _ _ _| | | _| _ _| |
%e . | | _ _ _ _ _| | 18 _ _| | | _ _ _ _ _ _ _ _
%e . | | | _ _ _ _ _| | | 39 _| _ _| | _ _ _ _ _ _ _|
%e . | | | | _ _ _ _| | _ _| _| | |
%e . | | | | | _ _ _ _| | _| 12 _| |
%e . | | | | | | _ _ _| | |_ _| _ _ _ _ _ _
%e . | | | | | | | _ _ _ _| 12 _ _| | _ _ _ _ _|
%e . | | | | | | | | _ _ _| | 9 _| |
%e . | | | | | | | | | _ _ _| 9 _|_ _|
%e . | | | | | | | | | | _ _| | _ _ _ _
%e . | | | | | | | | | | | _ _| 12 _| _ _ _|
%e . | | | | | | | | | | | | _| |
%e . | | | | | | | | | | | | | _ _|
%e . | | | | | | | | | | | | | | 3 _ _
%e . | | | | | | | | | | | | | | | _|
%e . |_| |_| |_| |_| |_| |_| |_| |_|
%e .
%e For n = 7 we have that 4*7-2 = 26 and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] and the 25th row of A237593 is [13, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 13] therefore between both Dyck paths there are two regions (or parts) of sizes [21, 21], so row 7 is [21, 21].
%e The sum of divisors of 26 is 1 + 2 + 13 + 26 = A000203(26) = 42. On the other hand the sum of the parts of the symmetric representation of sigma(26) is 21 + 21 = 42, equaling the sum of divisors of 26.
%Y Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239052, A239660, A239931, A239933, A239934, A244050, A245092, A262626.
%K nonn,tabf,more
%O 1,1
%A _Omar E. Pol_, Mar 29 2014
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