A143809 Eigentriangle of the Mobius transform, (A054525).

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

Offset: 1

Author

Gary W. Adamson, Sep 01 2008

Keywords

• tabl
• sign

Comments

The eigentriangle of the Mobius transform may be defined by the operation consisting of the termwise product of A054525 row terms and the first n terms of A007554, where A007554: (1, 1, 0, -1, -2, -3, -3,...) = the eigensequence of A054525.

This triangle has the following properties:

Sum of n-th row terms = rightmost term of next row.

Right border = A007554, the eigensequence of the Mobius transform.

Row sums = A007554 shifted one place to the left: (1, 0, -1, -2, -3,...).

Left border = mu(n), A008683.

A054525 = the Mobius transform and A007554 = the eigensequence of A054525.

Examples

			First few rows of the triangle:
   1;
  -1,  1;
  -1,  0, 0;
   0, -1, 0, -1;
  -1,  0, 0,  0, -2;
   1, -1, 0,  0,  0, -3;
  -1,  0, 0,  0,  0,  0, -3;
   0,  0, 0,  1,  0,  0,  0, -4;
   0,  0, 0,  0,  0,  0,  0,  0, -3;
   1, -1, 0,  0,  2,  0,  0,  0,  0, -3;
   ...
Row 6 = (1, -1, 0, 0, 0, -3) = termwise product of row 6 of the Mobius transform (1, -1, -1, 0, 0, 1) and the first 6 terms of A007554, (the eigensequence of the Mobius transform): (1, 1, 0, -1, -2, -3).
		

Crossrefs

Cf. A008683, A007554, A054525.

Formula

Triangle read by rows, A054525 * (A007554 * 0^(n-k)); 1<=k<=n