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%I #11 Dec 30 2023 23:43:52
%S 4,2,4,4,12,4,4,4,12,14,20,6,4,4,12,12,28,24,28,8,4,4,12,12,24,30,44,
%T 34,36,10,4,4,12,12,24,24,48,48,60,44,44,12,4,4,12,12,24,24,40,50,72,
%U 66,76,54,52,14,4,4,12,12,24,24,40,40,72,76,96,84,92
%N Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = 2n+1-k, where (x,y,z) is a permutation of three distinct numbers 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)!. The limiting row, (4, 4, 12, 12, 24, 24, 40, 40, ...) consists of repeated terms of (A046092(n+1)) = (4, 12, 24, 40, ...).
%e Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = 2n+1-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 (4,2), representing four 2s and two 3s.
%e First eight rows:
%e 4 2
%e 4 4 12 4
%e 4 4 12 14 20 6
%e 4 4 12 12 28 24 28 8
%e 4 4 12 12 24 30 44 34 36 10
%e 4 4 12 12 24 24 48 48 60 44 44 12
%e 4 4 12 12 24 24 40 50 72 66 76 54 52 14
%e 4 4 12 12 24 24 40 40 72 76 96 84 92 64 60 16
%t t[n_] := t[n] = Permutations[-1 + Range[n + 1], {3}];
%t a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == 2n+1-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, A046092, A368435, A368436.
%K nonn,tabf
%O 1,1
%A _Clark Kimberling_, Dec 25 2023
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