%I #33 Oct 07 2017 22:21:18
%S 1,1,1,3,1,0,1,5,1,3,0,1,0,7,1,0,0,1,3,9,1,0,5,0,1,0,0,11,1,3,0,0,1,0,
%T 0,13,1,0,7,0,1,3,5,0,15,1,0,0,0,0,1,0,0,0,17,1,3,0,9,0,1,0,0,0,19,1,
%U 0,5,0,0,1,3,0,7,0,21,1,0,0,0,11,0,1,0,0,0,0,23,1,3,0,0,0,0,1,0,5,0,0,25
%N Irregular triangle read by rows which arises from a diagram which is similar to the diagram of A261699, but here the even-indexed zig-zag lines are located on the right-hand side of the vertical axis of the diagram.
%C In the diagram we have that:
%C The number of horizontal line segments in the n-th row of the structure equals A001227(n), the number of partitions of n into consecutive parts.
%C The number of horizontal line segments in the left-hand part of the n-th row equals A082647, the number of partitions of n into an odd number of consecutive parts.
%C The number of horizontal line segments in the right-hand part of the n-th row equals A131576, the number of partitions of n into an even number of consecutive parts.
%C The diagram explains the unusual ordering of the terms in the triangle A261699, in which the even-indexed zig-zag lines are located on the left-hand side of the vertical axis of the diagram.
%e Triangle begins:
%e 1;
%e 1;
%e 1, 3;
%e 1, 0,
%e 1, 5;
%e 1, 3, 0;
%e 1, 0, 7;
%e 1, 0, 0;
%e 1, 3, 9;
%e 1, 0, 5, 0;
%e 1, 0, 0, 11;
%e 1, 3, 0, 0;
%e 1, 0, 0, 13;
%e 1, 0, 7, 0;
%e 1, 3, 5, 0, 15;
%e 1, 0, 0, 0, 0;
%e 1, 0, 0, 0, 17;
%e 1, 3, 0, 9, 0;
%e 1, 0, 0, 0, 19;
%e 1, 0, 5, 0, 0;
%e 1, 3, 0, 7, 0, 21;
%e ...
%e Illustration of initial terms of the diagram:
%e Row _
%e 1 _|1|
%e 2 _|1 |_
%e 3 _|1 |3|
%e 4 _|1 |0|_
%e 5 _|1 _| 5|
%e 6 _|1 |3| 0|_
%e 7 _|1 |0| 7|
%e 8 _|1 _|0| 0|_
%e 9 _|1 |3 |_ 9|
%e 10 _|1 |0 |5| 0|_
%e 11 _|1 _|0 |0| 11|
%e 12 _|1 |3 |0| 0|_
%e 13 _|1 |0 |0|_ 13|
%e 14 _|1 _|0 _| 7| 0|_
%e 15 _|1 |3 |5| 0| 15|
%e 16 _|1 |0 |0| 0| 0|_
%e 17 _|1 _|0 |0| 0|_ 17|
%e 18 _|1 |3 |0| 9| 0|_
%e 19 _|1 |0 _|0| 0| 19|
%e 20 _|1 _|0 |5 |_ 0| 0|_
%e 21 |1 |3 |0 |7| 0| 21|
%e ...
%e (Compare with the diagram of A261699.)
%Y Positive terms give A182469.
%Y Row n has length A003056(n).
%Y The sum of row n is A000593(n).
%Y Column k starts in row A000217(k).
%Y The number of positive terms in row n is A001227(n).
%Y Cf. A082647, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A259176, A259177, A261699, A279820.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Apr 21 2017
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