A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.

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.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

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

Cf. A144112, A185781, A185782, A185783.

Sequence in context: A347968 A197258 A211783 * A051672 A156308 A325004

Adjacent sequences: A185777 A185778 A185779 * A185781 A185782 A185783

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 03 2011

Status

approved