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A368520 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 <= z.

%I A368520 #4 Jan 25 2024 08:07:53
%S A368520 1,2,2,2,3,4,7,2,2,4,6,12,8,6,2,2,5,8,17,14,15,6,6,2,2,6,10,22,20,24,
%T A368520 16,12,6,6,2,2,7,12,27,26,33,26,25,12,12,6,6,2,2,8,14,32,32,42,36,38,
%U A368520 26,20,12,12,6,6,2,2,9,16,37,38,51,46,51,40,37,20,20
%N A368520 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 <=  z.
%C A368520 Row n consists of 2n-1 positive integers.
%e A368520 First seven rows:
%e A368520  1
%e A368520  2   2   2
%e A368520  3   4   7   2   2
%e A368520  4   6  12   8   6   2   2
%e A368520  5   8  17  14  15   6   6   2   2
%e A368520  6  10  22  20  24  16  12   6   6  2  2
%e A368520  7  12  27  26  33  26  25  12  12  6  6  2  2
%e A368520 For n=2, there are 6 triples (x,y,z) having x <= z:
%e A368520 111:  |x-y| + |y-z| = 0
%e A368520 112:  |x-y| + |y-z| = 1
%e A368520 121:  |x-y| + |y-z| = 2
%e A368520 122:  |x-y| + |y-z| = 1
%e A368520 212:  |x-y| + |y-z| = 2
%e A368520 222:  |x-y| + |y-z| = 0
%e A368520 so that row 1 of the array is (2,2,2), representing two 0s, two 1s, and two 2s.
%t A368520 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368520 t[n_] := t[n] = Select[t1[n], #[[1]] <= #[[3]] &];
%t A368520 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368520 u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
%t A368520 v = Flatten[u]  (* sequence *)
%t A368520 Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}]]  (* array *)
%Y A368520 Cf. A002411 (row sums), A110660 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368521, A368522.
%K A368520 nonn,tabf
%O A368520 1,2
%A A368520 _Clark Kimberling_, Jan 22 2024