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%I #10 Dec 30 2023 23:42:00
%S 2,4,4,12,4,4,6,20,14,12,4,4,8,28,24,28,12,12,4,4,10,36,34,44,30,24,
%T 12,12,4,4,12,44,44,60,48,48,24,24,12,12,4,4,14,52,54,76,66,72,50,40,
%U 24,24,12,12,4,4,16,60,64,92,84,96,76,72,40,40,24,24,12
%N Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where (x,y,z) is a permutation of three distinct numbers x,y,z taken from {0,1,...,n}, for n >= 2, k >= 2.
%C Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!.
%e Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = k, for k = 2,3, are as follows:
%e 012: |0-1| + |1-2| = 2
%e 021: |0-2| + |2-1| = 3
%e 102: |1-0| + |0-2| = 3
%e 120: |1-2| + |2-0| = 3
%e 201: |2-0| + |0-1| = 3
%e 210: |2-1| + |1-0| = 2
%e so that row 1 of the array is (2,4), representing two 2s and four 3s.
%e First eight rows:
%e 2 4
%e 4 12 4 4
%e 6 20 14 12 4 4
%e 8 28 24 28 12 12 4 4
%e 10 36 34 44 30 24 12 12 4 4
%e 12 44 44 60 48 48 24 24 12 12 4 4
%e 14 52 54 76 66 72 50 40 24 24 12 12 4 4
%e 16 60 64 92 84 96 76 72 40 40 24 24 12 12 4 4
%t t[n_] := t[n] = Permutations[-1 + Range[n + 1], {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, 2, 2 n - 1}];
%t v = Flatten[u] (* sequence *)
%t Column[u] (* array *)
%Y Cf. A007531, A368434, A368437 (reversed rows).
%K nonn,tabf
%O 1,1
%A _Clark Kimberling_, Dec 25 2023
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