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A185780
Array T(n,k) = k*(n*k-n+1), by antidiagonals.
4
1, 4, 1, 9, 6, 1, 16, 15, 8, 1, 25, 28, 21, 10, 1, 36, 45, 40, 27, 12, 1, 49, 66, 65, 52, 33, 14, 1, 64, 91, 96, 85, 64, 39, 16, 1, 81, 120, 133, 126, 105, 76, 45, 18, 1, 100, 153, 176, 175, 156, 125, 88, 51, 20, 1, 121, 190, 225, 232, 217, 186, 145, 100, 57, 22, 1, 144, 231, 280, 297, 288, 259, 216, 165, 112, 63, 24, 1, 169, 276, 341, 370, 369, 344, 301, 246, 185, 124, 69, 26, 1, 196, 325, 408, 451, 460, 441, 400, 343, 276, 205, 136, 75, 28, 1
OFFSET
1,2
COMMENTS
This is the accumulation array of A185781, the weight array of A185782, and second weight array of A185783. See A144112 for definitions of accumulation array and weight array.
FORMULA
T(n,k) = k*(n*k - n + 1), k>=1, n>=1.
EXAMPLE
Northwest corner:
1....4....9....16....25....36
1....6....15...28....45....66
1....8....21...40....65....96
1....10...27...52....85....126
MATHEMATICA
(* This code yields arrays A185780, A185781, and A185782. *)
f[n_, 0]:=0; f[0, k_]:=0; (* Used to make weight array A185782 *)
f[n_, k_]:=k(n*k-n+1);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] (* this array *)
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}]; (* acc array of {f(n, k)} *)
FullSimplify[s[n, k]]
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]] (* A185781 *)
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
w[m_, n_]:=f[m, n]+f[m-1, n-1]-f[m, n-1]-f[m-1, n]/; Or[m>0, n>0];
TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 15}]] (* array A185782 *)
Table[w[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* seq A185782 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 03 2011
STATUS
approved