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A285574 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.

%I A285574 #33 Oct 07 2017 22:21:18
%S A285574 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 A285574 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 A285574 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 A285574 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 A285574 In the diagram we have that:
%C A285574 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 A285574 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 A285574 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 A285574 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 A285574 Triangle begins:
%e A285574 1;
%e A285574 1;
%e A285574 1, 3;
%e A285574 1, 0,
%e A285574 1, 5;
%e A285574 1, 3, 0;
%e A285574 1, 0, 7;
%e A285574 1, 0, 0;
%e A285574 1, 3, 9;
%e A285574 1, 0, 5,  0;
%e A285574 1, 0, 0, 11;
%e A285574 1, 3, 0,  0;
%e A285574 1, 0, 0, 13;
%e A285574 1, 0, 7,  0;
%e A285574 1, 3, 5,  0, 15;
%e A285574 1, 0, 0,  0,  0;
%e A285574 1, 0, 0,  0, 17;
%e A285574 1, 3, 0,  9,  0;
%e A285574 1, 0, 0,  0, 19;
%e A285574 1, 0, 5,  0,  0;
%e A285574 1, 3, 0,  7,  0, 21;
%e A285574 ...
%e A285574 Illustration of initial terms of the diagram:
%e A285574 Row                                           _
%e A285574 1                                           _|1|
%e A285574 2                                         _|1  |_
%e A285574 3                                       _|1    |3|
%e A285574 4                                     _|1      |0|_
%e A285574 5                                   _|1       _|  5|
%e A285574 6                                 _|1        |3|  0|_
%e A285574 7                               _|1          |0|    7|
%e A285574 8                             _|1           _|0|    0|_
%e A285574 9                           _|1            |3  |_     9|
%e A285574 10                        _|1              |0  |5|    0|_
%e A285574 11                      _|1               _|0  |0|     11|
%e A285574 12                    _|1                |3    |0|      0|_
%e A285574 13                  _|1                  |0    |0|_      13|
%e A285574 14                _|1                   _|0   _|  7|      0|_
%e A285574 15              _|1                    |3    |5|  0|       15|
%e A285574 16            _|1                      |0    |0|  0|        0|_
%e A285574 17          _|1                       _|0    |0|  0|_        17|
%e A285574 18        _|1                        |3      |0|    9|        0|_
%e A285574 19      _|1                          |0     _|0|    0|         19|
%e A285574 20    _|1                           _|0    |5  |_   0|          0|_
%e A285574 21   |1                            |3      |0  |7|  0|           21|
%e A285574 ...
%e A285574 (Compare with the diagram of A261699.)
%Y A285574 Positive terms give A182469.
%Y A285574 Row n has length A003056(n).
%Y A285574 The sum of row n is A000593(n).
%Y A285574 Column k starts in row A000217(k).
%Y A285574 The number of positive terms in row n is A001227(n).
%Y A285574 Cf. A082647, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A259176, A259177, A261699, A279820.
%K A285574 nonn,tabf
%O A285574 1,4
%A A285574 _Omar E. Pol_, Apr 21 2017