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%I #70 Nov 21 2021 07:39:11
%S 0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,1,1,0,0,0,0,1,1,0,0,0,1,1,0,1,0,1,0,0,
%T 1,1,0,1,1,0,0,0,0,1,0,0,1,1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,1,
%U 1,0,0,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1
%N Irregular triangle read by rows: T(n,k) = 1 iff there is no partition of n into exactly k consecutive parts, for n >= 1, 1 <= k <= A003056(n).
%C The connection with A196020 is as follow: this sequence --> A237048 --> A235791 --> A236104 --> A196020.
%F T(n,k) = 1 - A237048(n,k).
%e Triangle begins (rows 1..28):
%e 0;
%e 0;
%e 0, 0;
%e 0, 1;
%e 0, 0;
%e 0, 1, 0;
%e 0, 0, 1;
%e 0, 1, 1;
%e 0, 0, 0;
%e 0, 1, 1, 0;
%e 0, 0, 1, 1;
%e 0, 1, 0, 1;
%e 0, 0, 1, 1;
%e 0, 1, 1, 0;
%e 0, 0, 0, 1, 0;
%e 0, 1, 1, 1, 1;
%e 0, 0, 1, 1, 1;
%e 0, 1, 0, 0, 1;
%e 0, 0, 1, 1, 1;
%e 0, 1, 1, 1, 0;
%e 0, 0, 0, 1, 1, 0;
%e 0, 1, 1, 0, 1, 1;
%e 0, 0, 1, 1, 1, 1;
%e 0, 1, 0, 1, 1, 1;
%e 0, 0, 1, 1, 0, 1;
%e 0, 1, 1, 0, 1, 1;
%e 0, 0, 0, 1, 1, 0;
%e 0, 1, 1, 1, 1, 1, 0;
%e ...
%e For n = 15 the partitions of 15 into consecutive parts are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. We can see that a partition of 15 into four consecutive parts does not exist, hence there is a "nonexistence" of such partition, so T(15,4) = 1.
%e Illustration of initial terms:
%e Row _
%e 1 _|0|
%e 2 _|0 _|
%e 3 _|0 |0|
%e 4 _|0 _|1|
%e 5 _|0 |0 _|
%e 6 _|0 _|1|0|
%e 7 _|0 |0 |1|
%e 8 _|0 _|1 _|1|
%e 9 _|0 |0 |0 _|
%e 10 _|0 _|1 |1|0|
%e 11 _|0 |0 _|1|1|
%e 12 _|0 _|1 |0 |1|
%e 13 _|0 |0 |1 _|1|
%e 14 _|0 _|1 _|1|0 _|
%e 15 _|0 |0 |0 |1|0|
%e 16 _|0 _|1 |1 |1|1|
%e 17 _|0 |0 _|1 _|1|1|
%e 18 _|0 _|1 |0 |0 |1|
%e 19 _|0 |0 |1 |1 _|1|
%e 20 _|0 _|1 _|1 |1|0 _|
%e 21 _|0 |0 |0 _|1|1|0|
%e 22 _|0 _|1 |1 |0 |1|1|
%e 23 _|0 |0 _|1 |1 |1|1|
%e 24 _|0 _|1 |0 |1 _|1|1|
%e 25 _|0 |0 |1 _|1|0 |1|
%e 26 _|0 _|1 _|1 |0 |1 _|1|
%e 27 _|0 |0 |0 |1 |1|0 _|
%e 28 |0 |1 |1 |1 |1|1|0|
%e ...
%e For the connection between the above diagram and the partitions of n into consecutive parts see the diagrams of A286000 and A286001.
%Y Row sums give A238005.
%Y Row n has length A003056(n).
%Y Column k starts in row A000217(k).
%Y The number of zeros in row n equals A001227(n).
%Y Cf. A196020, A235791, A236104, A237048, A237593, A280850, A286000, A286001, A296508.
%K nonn,tabf
%O 1
%A _Omar E. Pol_, Sep 07 2021