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A339275 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the terms of A040000: 1, 2, 2, 2, ... interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

%I A339275 #60 Dec 11 2020 13:27:51
%S A339275 1,2,2,1,2,0,2,2,2,0,1,2,2,0,2,0,0,2,2,2,2,0,0,1,2,2,0,0,2,0,2,0,2,2,
%T A339275 0,0,2,0,0,2,2,2,2,0,1,2,0,0,0,0,2,2,0,0,0,2,0,2,2,0,2,2,0,0,0,2,0,0,
%U A339275 0,2,2,2,2,0,0,1,2,0,0,2,0,0,2,2,0,0,0,0,2,0,2,0,0,0,2,2,0,0,2,0,2
%N A339275 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the terms of A040000: 1, 2, 2, 2, ... interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.
%C A339275 T(n,k) is also the number of horizontal line segments in the n-th level of the k-th largest double-staircase of the diagram defined in A335616 (see example).
%C A339275 The partial sums of column k give the k-th column of A338721.
%e A339275 Triangle begins (rows 1..28):
%e A339275 1;
%e A339275 2;
%e A339275 2,  1;
%e A339275 2,  0;
%e A339275 2,  2;
%e A339275 2,  0,  1;
%e A339275 2,  2,  0;
%e A339275 2,  0,  0;
%e A339275 2,  2,  2;
%e A339275 2,  0,  0,  1;
%e A339275 2,  2,  0,  0;
%e A339275 2,  0,  2,  0;
%e A339275 2,  2,  0,  0;
%e A339275 2,  0,  0,  2;
%e A339275 2,  2,  2,  0,  1;
%e A339275 2,  0,  0,  0,  0;
%e A339275 2,  2,  0,  0,  0;
%e A339275 2,  0,  2,  2,  0;
%e A339275 2,  2,  0,  0,  0;
%e A339275 2,  0,  0,  0,  2;
%e A339275 2,  2,  2,  0,  0,  1;
%e A339275 2,  0,  0,  2,  0,  0;
%e A339275 2,  2,  0,  0,  0,  0;
%e A339275 2,  0,  2,  0,  0,  0;
%e A339275 2,  2,  0,  0,  2,  0;
%e A339275 2,  0,  0,  2,  0,  0;
%e A339275 2,  2,  2,  0,  0,  2;
%e A339275 2,  0,  0,  0,  0,  0,  1;
%e A339275 ...
%e A339275 For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
%e A339275 The first 15 levels of the structure looks like this:
%e A339275 .
%e A339275 Level                         "Double-staircases" diagram
%e A339275 n                                          _
%e A339275 1                                        _|1|_
%e A339275 2                                      _|1 _ 1|_
%e A339275 3                                    _|1  |1|  1|_
%e A339275 4                                  _|1   _| |_   1|_
%e A339275 5                                _|1    |1 _ 1|    1|_
%e A339275 6                              _|1     _| |1| |_     1|_
%e A339275 7                            _|1      |1  | |  1|      1|_
%e A339275 8                          _|1       _|  _| |_  |_       1|_
%e A339275 9                        _|1        |1  |1 _ 1|  1|        1|_
%e A339275 10                     _|1         _|   | |1| |   |_         1|_
%e A339275 11                   _|1          |1   _| | | |_   1|          1|_
%e A339275 12                 _|1           _|   |1  | |  1|   |_           1|_
%e A339275 13               _|1            |1    |  _| |_  |    1|            1|_
%e A339275 14             _|1             _|    _| |1 _ 1| |_    |_             1|_
%e A339275 15            |1              |1    |1  | |1| |  1|    1|              1|
%e A339275 .
%e A339275 For n = 15, in the 15th level of the diagram we have that the first largest double-staircase has two horizontal steps, the second double-staircase has two steps, the third double-staircase has two steps, there are no steps in the fourth double-stairce and the fifth double-staircase has only one step, so the 15th row of triangle is [2, 2, 2, 0, 1].
%Y A339275 Column 1 is A040000.
%Y A339275 Row sums give A335616.
%Y A339275 Row n has length A003056(n).
%Y A339275 Column k starts in row A000217(k).
%Y A339275 The number of positive terms in row n is A001227(n).
%Y A339275 Cf. A196020, A236104, A237048, A237270, A237591, A237593, A249351, A280850, A296508, A299484, A338721.
%K A339275 nonn,tabf
%O A339275 1,2
%A A339275 _Omar E. Pol_, Dec 01 2020