Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #60 Dec 11 2020 13:27:51
%S 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 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 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 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 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 The partial sums of column k give the k-th column of A338721.
%e Triangle begins (rows 1..28):
%e 1;
%e 2;
%e 2, 1;
%e 2, 0;
%e 2, 2;
%e 2, 0, 1;
%e 2, 2, 0;
%e 2, 0, 0;
%e 2, 2, 2;
%e 2, 0, 0, 1;
%e 2, 2, 0, 0;
%e 2, 0, 2, 0;
%e 2, 2, 0, 0;
%e 2, 0, 0, 2;
%e 2, 2, 2, 0, 1;
%e 2, 0, 0, 0, 0;
%e 2, 2, 0, 0, 0;
%e 2, 0, 2, 2, 0;
%e 2, 2, 0, 0, 0;
%e 2, 0, 0, 0, 2;
%e 2, 2, 2, 0, 0, 1;
%e 2, 0, 0, 2, 0, 0;
%e 2, 2, 0, 0, 0, 0;
%e 2, 0, 2, 0, 0, 0;
%e 2, 2, 0, 0, 2, 0;
%e 2, 0, 0, 2, 0, 0;
%e 2, 2, 2, 0, 0, 2;
%e 2, 0, 0, 0, 0, 0, 1;
%e ...
%e For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
%e The first 15 levels of the structure looks like this:
%e .
%e Level "Double-staircases" diagram
%e n _
%e 1 _|1|_
%e 2 _|1 _ 1|_
%e 3 _|1 |1| 1|_
%e 4 _|1 _| |_ 1|_
%e 5 _|1 |1 _ 1| 1|_
%e 6 _|1 _| |1| |_ 1|_
%e 7 _|1 |1 | | 1| 1|_
%e 8 _|1 _| _| |_ |_ 1|_
%e 9 _|1 |1 |1 _ 1| 1| 1|_
%e 10 _|1 _| | |1| | |_ 1|_
%e 11 _|1 |1 _| | | |_ 1| 1|_
%e 12 _|1 _| |1 | | 1| |_ 1|_
%e 13 _|1 |1 | _| |_ | 1| 1|_
%e 14 _|1 _| _| |1 _ 1| |_ |_ 1|_
%e 15 |1 |1 |1 | |1| | 1| 1| 1|
%e .
%e 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 Column 1 is A040000.
%Y Row sums give A335616.
%Y Row n has length A003056(n).
%Y Column k starts in row A000217(k).
%Y The number of positive terms in row n is A001227(n).
%Y Cf. A196020, A236104, A237048, A237270, A237591, A237593, A249351, A280850, A296508, A299484, A338721.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Dec 01 2020