%I #12 Jan 29 2024 11:01:34
%S 1,1,2,3,2,1,3,5,5,4,2,1,4,7,8,8,6,4,2,1,5,9,11,12,11,9,6,4,2,1,6,11,
%T 14,16,16,15,12,9,6,4,2,1,7,13,17,20,21,21,19,16,12,9,6,4,2,1,8,15,20,
%U 24,26,27,26,24,20,16,12,9,6,4,2,1,9,17,23,28
%N 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 Row n consists of 2n positive integers.
%e First six rows:
%e 1 1
%e 2 3 2 1
%e 3 5 5 4 2 1
%e 4 7 8 8 6 4 2 1
%e 5 9 11 12 11 9 6 4 2 1
%e 6 11 14 16 16 15 12 9 6 4 2 1
%e For n=3, there are 8 triples (x,y,z) having x < y and y >= z:
%e 121: |x-y| + |y-z| = 2
%e 122: |x-y| + |y-z| = 1
%e 131: |x-y| + |y-z| = 4
%e 132: |x-y| + |y-z| = 3
%e 133: |x-y| + |y-z| = 2
%e 231: |x-y| + |y-z| = 3
%e 232: |x-y| + |y-z| = 2
%e 233: |x-y| + |y-z| = 1
%e so row 1 of the array is (2,3,2,1), representing two 1s, three 2s, two 3s, and one 4.
%t t1[n_] := t1[n] = Tuples[Range[n], 3];
%t t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] >= #[[3]] &];
%t a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
%t v = Flatten[u] (* sequence *)
%t Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] ((* array *)
%Y Cf. A000027 (column 1), A007290 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368606, A368607, A368609.
%K nonn,tabf
%O 1,3
%A _Clark Kimberling_, Jan 22 2024