A127448 Triangle T(n,k) read by rows: matrix product A054525 * A127648.

1, -1, 2, -1, 0, 3, 0, -2, 0, 4, -1, 0, 0, 0, 5, 1, -2, -3, 0, 0, 6, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0, -4, 0, 0, 0, 8, 0, 0, -3, 0, 0, 0, 0, 0, 9, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, -4, 0, -6, 0, 0, 0, 0, 0, 12, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Jan 14 2007

Keywords

• tabl
• sign
• easy

Examples

			First few rows of the triangle are;
1;
-1, 2;
-1, 0, 3;
0, -2, 0, 4;
-1, 0, 0, 0, 5;
1, -2, -3, 0, 0, 6;
-1, 0, 0, 0, 0, 0, 7;
0, 0, 0, -4, 0, 0, 0, 8;
0, 0, -3, 0, 0, 0, 0, 0, 9;
1, -2, 0, 0,-5, 0, 0, 0, 0, 10;
...
		

Crossrefs

Cf. A000010, A008683, A051731.

Programs

Maple

A127648 := proc(n,k) if n = k then n; else 0 ; fi; end:
A054525 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi; end:
A127448 := proc(n,k) add( A054525(n,j)*A127648(j,k), j=k..n) ; end: seq(seq( A127448(n,k),k=1..n),n=1..15) ;

Formula

T(n,k) = sum _{j=k..n} A054525(n,j)*A127648(j,k) = k*A054525(n,k).

sum_{k=1..n} T(n,k) = A000010(n) (row sums).

T(n,1) = A008683(n).

Extensions

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

Corrected A-number typo in a formula - R. J. Mathar, Sep 17 2009

Corrected last example line by John Mason, Jan 07 2015