A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 12, 12, 4, 0, 0, 0, 32, 54, 32, 3, 0, 0, 0, 80, 216, 192, 30, 7, 0, 0, 0, 192, 810, 1024, 225, 84, 4, 0, 0, 0, 448, 2916, 5120, 1500, 756, 56, 12, 0, 0, 0, 1024, 10206, 24576, 9375, 6048, 588, 192, 12, 0

Offset: 0

Author

Alois P. Heinz, Jun 27 2015

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 1,  4,  12,   32,    80,    192,     448, ...
  0, 2, 12,  54,  216,   810,   2916,   10206, ...
  0, 4, 32, 192, 1024,  5120,  24576,  114688, ...
  0, 3, 30, 225, 1500,  9375,  56250,  328125, ...
  0, 7, 84, 756, 6048, 45360, 326592, 2286144, ...
  0, 4, 56, 588, 5488, 48020, 403368, 3294172, ...
		

Links

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

Crossrefs

Rows n=0+1,2,3,4,8 give: A000004, A001787, A212697, A018215, A230539.

Columns k=0,1 give: A000004, A258851.

Main diagonal gives A258846.

Cf. A000720.

Programs

Maple

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= (n, k)-> `if`(k=0, 0, k*n^(k-1)*d(n)):
seq(seq(A(n, h-n), n=0..h), h=0..14);

Formula

A(n,k) = A258851(n^k) = k * n^(k-1) * A258851(n).