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A368607 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 A368607 #8 Jan 29 2024 11:01:43
%S A368607 1,3,2,1,5,6,4,2,1,7,10,10,6,4,2,1,9,14,16,14,9,6,4,2,1,11,18,22,22,
%T A368607 19,12,9,6,4,2,1,13,22,28,30,29,24,16,12,9,6,4,2,1,15,26,34,38,39,36,
%U A368607 30,20,16,12,9,6,4,2,1,17,30,40,46,49,48,44,36,25
%N A368607 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 A368607 Row n consists of 2n-1 positive integers.
%e A368607 First six rows:
%e A368607  1
%e A368607  3    2   1
%e A368607  5    6   4   2  1
%e A368607  7   10  10   6  4   2   1
%e A368607  9   14  16  14  9   6   4  2  1
%e A368607  11  18  22  22  19  12  9  6  4  2  1
%e A368607 For n=3, there are 6 triples (x,y,z) having x != y and y < z:
%e A368607   123:  |x-y| + |y-z| = 2
%e A368607   212:  |x-y| + |y-z| = 2
%e A368607   213:  |x-y| + |y-z| = 3
%e A368607   312:  |x-y| + |y-z| = 3
%e A368607   313:  |x-y| + |y-z| = 4
%e A368607   323:  |x-y| + |y-z| = 2
%e A368607 so row 2 of the array is (3,2,1), representing three 2s, two 3s, and one 4.
%t A368607 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368607 t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] < #[[3]] &];
%t A368607 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368607 u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}];
%t A368607 v = Flatten[u] (* sequence *)
%t A368607 Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}]] (* array *)
%Y A368607 Cf. A005408 (column 1), A002411 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368609.
%K A368607 nonn,tabf
%O A368607 1,2
%A A368607 _Clark Kimberling_, Jan 25 2024