%I #93 Jan 18 2019 10:00:48
%S 1,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,1,1,0,0,1,1,0,1,0,1,
%T 1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0,1,1,
%U 1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0,0,1,1,1,0,1,1,0,0,0,0,1,1,0,1,0
%N Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by two, including the partition n, and the first element of column k is in row k^2.
%C T(n,k) is 0 or 1, so T(n,k) represents the "existence" of the mentioned partition: 1 = exists, 0 = does not exist.
%C Since the trivial partition n is counted, so T(n,1) = 1.
%C This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists 1's interleaved with k-1 zeros, and the first element of column k is in row k^2.
%C From _Omar E. Pol_, Jan 08 2019: (Start)
%C Theorem: Let T(n,k) be an irregular triangle read by rows in which column k lists 1's interleaved with k-1 zeros, and the first element of column k is where the row number equals the k-th (m+2)-gonal number, with n >= 1, k >= 1, m >= 0. T(n,k) is also the number of partitions of n into k consecutive parts that differ by m, including the partition n.
%C About the above theorem, this is the case for m = 2. For m = 1 see the triangle A237048, in which row sums give A001227. For m = 0 see the triangle A051731, in which row sums give A000005. Note that there are infinitely many triangles of this kind, with m >= 0. Also, every triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve. (End)
%e Triangle begins (rows 1..25):
%e 1;
%e 1;
%e 1;
%e 1, 1;
%e 1, 0;
%e 1, 1;
%e 1, 0;
%e 1, 1;
%e 1, 0, 1;
%e 1, 1, 0;
%e 1, 0, 0;
%e 1, 1, 1;
%e 1, 0, 0;
%e 1, 1, 0;
%e 1, 0, 1;
%e 1, 1, 0, 1;
%e 1, 0, 0, 0;
%e 1, 1, 1, 0;
%e 1, 0, 0, 0;
%e 1, 1, 0, 1;
%e 1, 0, 1, 0;
%e 1, 1, 0, 0;
%e 1, 0, 0, 0;
%e 1, 1, 1, 1;
%e 1, 0, 0, 0, 1;
%e ...
%e For n = 16 there are three partitions of 16 into consecutive parts that differ by two, including 16 as a partition. They are [16], [9, 7] and [7, 5, 3, 1]. The number of parts of these partitions are 1, 2 and 4 respectively, so the 16th row of the triangle is [1, 1, 0, 1].
%Y Row sums give A038548.
%Y Cf. A000005, A001227, A000290, A051731, A139600, A237048.
%K nonn,tabf
%O 1
%A _Omar E. Pol_, Apr 21 2018