A127173 T(n,k) = A007427(n/k) if k divides n, T(n,k) = 0 otherwise.

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

Offset: 1

Author

Gary W. Adamson, Jan 06 2007

Keywords

• tabl
• sign

Examples

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A008683.

Column 1 is A007427.

Cf. A054525, A007431, A007428.

Programs

PARI

\\ here b(n) is A007427(n).
b(n)={sumdiv(n, d, moebius(d) * moebius(n/d))}
T(n,k)={if(n%k==0, b(n/k), 0)} \\ Andrew Howroyd, Feb 20 2022

Formula

Square of A054525 as lower triangular matrix.

A007431(n) = Sum_{k=1, n} k*T(n,k).

A007428(n) = Sum_{k=1..n} mu(k)*T(n,k).

Extensions

Missing a(10)-a(14) and a(56) and beyond from Andrew Howroyd, Feb 20 2022