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

Author

Clark Kimberling, Feb 03 2011

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A144112, A185781, A185782, A185783.

Programs

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 *)

Formula

T(n,k) = k*(n*k - n + 1), k>=1, n>=1.