A166278 Square array A(n,k), n,k>=0, read by antidiagonals: A(n,k) is the total element sum of the k-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 6, 10, 10, 5, 0, 1, 8, 19, 20, 15, 6, 0, 1, 12, 33, 46, 35, 21, 7, 0, 1, 16, 63, 100, 94, 56, 28, 8, 0, 1, 24, 111, 220, 242, 172, 84, 36, 9, 0, 1, 32, 201, 488, 633, 514, 290, 120, 45, 10, 0, 1, 48, 369, 1104, 1643, 1518, 984, 460, 165, 55, 11

Offset: 0

Author

Alois P. Heinz, Oct 10 2009

Keywords

• easy
• nonn
• tabl

Examples

			A(3,4) = 33, because f([1,1,1]) = [1,2,3], (f^2)([1,1,1]) = [3,3,4], (f^3)([1,1,1]) = [4,6,9], (f^4)([1,1,1]) = [9,12,12], and 9+12+12 = 33.
Square array A(n,k) begins:
  0,  0,  0,  0,   0,   0, ...
  1,  1,  1,  1,   1,   1, ...
  2,  3,  4,  6,   8,  12, ...
  3,  6, 10, 19,  33,  63, ...
  4, 10, 20, 46, 100, 220, ...
  5, 15, 35, 94, 242, 633, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-3 give: A001477, A000217, A000292, A070893.

Rows n=0-2 give: A000004, A000012, A029744(k+2).

Main diagonal gives A261490.

Programs

Maple

f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
A:= (n, k)-> add(i, i=(f@@k)([1$n])):
seq(seq(A(n, d-n), n=0..d), d=0..15);

Mathematica

f[L_List] := f[L] = Sort[Reverse[Sort[L]]*Range[Length[L]]];
A[0, ] = 0; A[n, 0] := n; A[n_, k_] := Total[Nest[f, Range[n], k-1]];
Table[A[n, k-n], {k, 0, 15}, {n, 0, k}] // Flatten (* Jean-François Alcover, Jun 07 2016 *)