A128408 -id:A128408 - cp's OEIS frontend

A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

1, 0, -1, -1, -1, -1, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, 0, -1, 1, -2, 0, 0, 0, -1, 1, -1, -2, 0, 0, 0, -1, 1, -1, 0, -2, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -3, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left border = the Mertens sequence: A002321: (1, 0, -1, -1, -2, ...).
Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1, ...).
Row sums = A062563: (1, -1, -3, -3, -5, -1, -3, ...).

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

Cf. A002321, A008683, A062563, A128407, A128408.

Previous a(51) = 0 removed and more terms from Georg Fischer, Jun 08 2023

A128410 A128408 * A000012.

Original entry on oeis.org

1, -2, -1, -2, -1, -1, 0, 0, 0, 0, -2, -1, -1, -1, -1, 4, 3, 2, 1, 1, 1, -2, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left border = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). Row sums = A063441: (1, -3, -4, 0, -6, 12, ...) A128409 = A000012 * A128408.

			First few rows of the triangle:
   1;
  -2, -1;
  -2, -1, -1;
   0,  0,  0,  0;
  -2, -1, -1, -1, -1;
   4,  3,  2,  1,  1,  1;
  -2, -1, -1, -1, -1, -1, -1;
  ...

Cf. A128407, A128408, A128409, A008966, A051731, A063441.

A128408 * A000012 as infinite lower triangular matrices. Partial row sums of A128408 starting from the right.

A128407 Triangle read by rows: T(n,n) = mobius(n) on the diagonal, zero elsewhere.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 01 2007

A128407 * [1,2,3,...] = n*mu(n), A055615: (1, -2, -3, 0, -5, 6, -7, 0, ...). A051731 * A128407 = A054524. A128407 * A051731 = A128408.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051731, A128408, A008683 (row sums and diagonal), A054524.

Table[PadLeft[{MoebiusMu[n]},n],{n,10}]//Flatten (* Harvey P. Dale, Dec 29 2021 *)

Infinite lower triangular matrix with mu(n), A008683, as the right border and the rest zeros.

Name improved . - R. J. Mathar, Aug 03 2015

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A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

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A128410 A128408 * A000012.

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A128407 Triangle read by rows: T(n,n) = mobius(n) on the diagonal, zero elsewhere.

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