A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.

1, 2, -1, 3, 0, -1, 4, -2, 0, 0, 5, 0, 0, 0, -1, 6, -3, -2, 0, 0, 1, 7, 0, 0, 0, 0, 0, -1, 8, -4, 0, 0, 0, 0, 0, 0, 9, 0, -3, 0, 0, 0, 0, 0, 0, 10, -5, 0, 0, -2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Aug 01 2008

Keywords

• tabl
• sign

Comments

Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".

Examples

			First few rows of the triangle are:
   1;
   2, -1;
   3,  0, -1;
   4, -2,  0,  0;
   5,  0,  0,  0, -1;
   6, -3, -2,  0,  0,  1;
   7,  0,  0,  0,  0,  0, -1;
   8, -4,  0,  0,  0,  0,  0,  0;
   9,  0, -3,  0,  0,  0,  0,  0,  0;
  10, -5,  0,  0, -2,  0,  0,  0,  0,  1;
  11,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
  12, -6, -4,  0,  0,  2,  0,  0,  0,  0,  0,  0;
  13,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
  14, -7,  0,  0,  0,  0, -2,  0,  0,  0,  0,  0,  0,  1;
  ...
Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).
		

References

• Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000010 (row sums), A008683, A126988, A128407.

Programs

Magma

A143239:= func< n,k | (n mod k) eq 0 select MoebiusMu(k)*(n/k) else 0 >;
[A143239(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024

Mathematica

A143239[n_, k_]:= If[Mod[n,k]==0, MoebiusMu[k]*(n/k), 0];
Table[A143239[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

SageMath

def A143239(n,k): return moebius(k)*(n//k) if (n%k)==0 else 0
flatten([[A143239(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Sep 12 2024

Formula

Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n, otherwise T(n,k) = 0 if k is not a divisor of n.

Equals A126988 * A128407.