A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

1, 0, 1, -1, 1, 1, -1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, -2, -1, 0, 1, 1, 1, 1, -2, -1, 0, 0, 1, 1, 1, 1, -2, -1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -1, 0, 0, 1, 1, 1, 1, 1, -2, -2, -1, 0, 0, 1, 1, 1, 1, 1, 1, -2, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gary W. Adamson, Oct 31 2007

Keywords

• tabl
• sign

Comments

Row sums = 1.

Left border = A002321, the Mertens function.

A134541 * [1,2,3,...] = A002088: (1, 2, 4, 6, 10, 12, 18, 22, ...).

Examples

			First few rows of the triangle:
   1;
   0,  1;
  -1,  1,  1;
  -1,  0,  1, 1;
  -2,  0,  1, 1, 1;
  -1, -1,  0, 1, 1, 1;
  -2, -1,  0, 1, 1, 1, 1;
  -2, -1,  0, 0, 1, 1, 1, 1;
  -2, -1, -1, 0, 1, 1, 1, 1, 1;
  -1, -2, -1, 0, 0, 1, 1, 1, 1, 1;
  ...
		

Crossrefs

Cf. A000012, A054525, A002321, A002088, A134541.

Matrix inverse of A176702. - Mats Granvik, Apr 24 2010

Programs

Mathematica

Clear[t, s, n, k, z, x]; z = 1; nn = 10; t[n_, k_] := t[n, k] = If[n >= k, If[k == 1, 1 - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], t[Floor[n/k], 1]], 0]; Flatten[Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jul 22 2012 *) (* updated Mats Granvik, Apr 10 2016 *)

Formula

Recurrence: T(n, k) = If n >= k then If k = 1 then 1 - Sum_{i=1..n-1} T(n, k + i)/(i + 1)^(s - 1) else T(floor(n/k) else 1)) else 0). - Mats Granvik, Apr 17 2016

Extensions

More terms from Amiram Eldar, Jun 09 2024