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.

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, 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, 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

Offset: 1

Author

Omar E. Pol, Dec 01 2020

Keywords

• nonn
• tabf

Comments

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).

The partial sums of column k give the k-th column of A338721.

Examples

			Triangle begins (rows 1..28):
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,  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,  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,  0,  0,  2,  0,  0;
2,  2,  2,  0,  0,  2;
2,  0,  0,  0,  0,  0,  1;
...
For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
The first 15 levels of the structure looks like this:
.
Level                         "Double-staircases" diagram
n                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
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].
		

Crossrefs

Column 1 is A040000.

Row sums give A335616.

Row n has length A003056(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n).

Cf. A196020, A236104, A237048, A237270, A237591, A237593, A249351, A280850, A296508, A299484, A338721.