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A368158 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.

%I A368158 #7 Jan 22 2024 00:02:26
%S A368158 2,3,1,3,6,6,2,1,4,9,11,9,4,2,1,5,12,16,16,13,6,4,2,1,6,15,21,23,22,
%T A368158 17,9,6,4,2,1,7,18,26,30,31,28,22,12,9,6,4,2,1,8,21,31,37,40,39,35,27,
%U A368158 16,12,9,6,4,2,1,9,24,36,44,49,50,48,42,33,20,16
%N A368158 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.
%C A368158 Row n consists of 2n+1 positive integers.
%e A368158 First six rows:
%e A368158   2   3   1
%e A368158   3   6   6   2   1
%e A368158   4   9  11   9   4   2   1
%e A368158   5  12  16  16  13   6   4   2  1
%e A368158   6  15  21  23  22  17   9   6  4  2  1
%e A368158   7  18  26  30  31  28  22  12  9  6  4  2  1
%e A368158 For n=2, there are 6 triples (x,y,z) having x <= y:
%e A368158   111:  |x-y| + |y-z| = 0
%e A368158   112:  |x-y| + |y-z| = 1
%e A368158   121:  |x-y| + |y-z| = 2
%e A368158   122:  |x-y| + |y-z| = 1
%e A368158   221:  |x-y| + |y-z| = 1
%e A368158   222:  |x-y| + |y-z| = 0,
%e A368158 so row 1 of the array is (2,3,1), representing two 0s, three 1s, and one 1.
%t A368158 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368158 t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] &];
%t A368158 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368158 u = Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}];
%t A368158 v = Flatten[u]  (* sequence *)
%t A368158 Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}]]  (* array *)
%Y A368158 Cf. A002411 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368519, A368520, A368521, A368522.
%K A368158 nonn,tabf
%O A368158 1,1
%A A368158 _Clark Kimberling_, Jan 20 2024