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A368606 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x <= y and y >= z.

%I A368606 #10 Jan 29 2024 11:01:39
%S A368606 1,2,2,1,3,4,4,2,1,4,6,7,6,4,2,1,5,8,10,10,9,6,4,2,1,6,10,13,14,14,12,
%T A368606 9,6,4,2,1,7,12,16,18,19,18,16,12,9,6,4,2,1,8,14,19,22,24,24,23,20,16,
%U A368606 12,9,6,4,2,1,9,16,22,26,29,30,30,28,25,20,16
%N A368606 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x <= y and y >= z.
%C A368606 Row n consists of 2n-1 positive integers.
%e A368606 First six rows:
%e A368606   1
%e A368606   2    2    1
%e A368606   3    4    4    2    1
%e A368606   4    6    7    6    4    2   1
%e A368606   5    8   10   10    9    6   4   2   1
%e A368606   6   10   13   14   14   12   9   6   4   2   1
%e A368606 For n=2, there are 5 triples (x,y,z) having x <= y and y >= z:
%e A368606   111:  |x-y| + |y-z| = 0
%e A368606   121:  |x-y| + |y-z| = 2
%e A368606   122:  |x-y| + |y-z| = 1
%e A368606   221:  |x-y| + |y-z| = 1
%e A368606   222:  |x-y| + |y-z| = 0
%e A368606 so row 2 of the array is (2,2,1), representing two 0s, two 1s, and one 3.
%t A368606 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368606 t[n_] := t[n] = Select[t1[n], #[[1]] <= #[[2]] >= #[[3]] &];
%t A368606 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368606 u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
%t A368606 v = Flatten[u]  (* sequence *)
%t A368606 Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}]]  (* array *)
%Y A368606 Cf. A000027 (column 1), A000330 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368607, A368609.
%K A368606 nonn,tabf
%O A368606 1,2
%A A368606 _Clark Kimberling_, Jan 22 2024