OFFSET
1,1
COMMENTS
Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!.
EXAMPLE
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:
012: |0-1| + |1-2| = 2
021: |0-2| + |2-1| = 3
102: |1-0| + |0-2| = 3
120: |1-2| + |2-0| = 3
201: |2-0| + |0-1| = 3
210: |2-1| + |1-0| = 2
so that row 1 of the array is (2,4), representing two 2s and four 3s.
First eight rows:
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 12 12 4 4
12 44 44 60 48 48 24 24 12 12 4 4
14 52 54 76 66 72 50 40 24 24 12 12 4 4
16 60 64 92 84 96 76 72 40 40 24 24 12 12 4 4
MATHEMATICA
t[n_] := t[n] = Permutations[-1 + Range[n + 1], {3}];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 1}];
v = Flatten[u] (* sequence *)
Column[u] (* array *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 25 2023
STATUS
approved