A127569 Triangle read by rows: product of the Mobius matrix A054525 by the diagonal matrix diag(A000203(n)).

1, -1, 3, -1, 0, 4, 0, -3, 0, 7, -1, 0, 0, 0, 6, 1, -3, -4, 0, 0, 12, -1, 0, 0, 0, 0, 0, 8, 0, 0, 0, -7, 0, 0, 0, 15, 0, 0, -4, 0, 0, 0, 0, 0, 13, 1, -3, 0, 0, -6, 0, 0, 0, 0, 18, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 3, 0, -7, 0, -12, 0, 0, 0, 0, 0, 28, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 1, -3, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 24, 1, 0, -4, 0, -6, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Jan 19 2007

Keywords

• tabl
• sign

Comments

Left column = mu(n), A008683; right border = sigma(n), A000203; row sums = n.

The definition of Mobius transform is extended to matrices here in the sense of "left multiplication by the Mobius matrix A054525". - R. J. Mathar, Oct 02 2007

Examples

			First few rows of the triangle are:
1;
-1, 3;
-1, 0, 4;
0, -3, 0, 7;
-1, 0, 0, 0, 6;
1, -3, -4, 0, 0, 12;
...
		

Crossrefs

Cf. A054525, A000203, A008683.

Programs

Maple

A000203T := proc(n,k) if n = k then numtheory[sigma](n) ; else 0 ; fi ; end: A054525 := proc(n,k) if n < 1 or k > n or n mod k <> 0 then 0; else numtheory[mobius](n/k) ; fi ; end: A127569 := proc(n,k) add(A054525(n,i)*A000203T(i,k),i=1..n) ; end: for n from 1 to 15 do for k from 1 to n do printf("%a, ",A127569(n,k)) ; od: od: # R. J. Mathar, Oct 02 2007

Formula

T(n,k)=A054525(n,k)*A000203(k). - R. J. Mathar, Oct 02 2007

Extensions

Missing comma corrected by Naruto Canada, Aug 26 2007

More terms from R. J. Mathar, Oct 02 2007