A206706 Triangle read by rows, T(n,k) n>=0, 0<=k<=n; T(0,0) = -1 and for n > 0 T(n,k) = moebius(n,k+1) - moebius(n,k) where moebius(n,k) = mu(floor(n/k)) if k<>0 and k divides n, 0 otherwise; mu=A008683.

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

Offset: 0

Author

Peter Luschny, Feb 11 2012

Keywords

• sign
• tabl

Comments

This is a variant of Paul D. Hanna's A123706 which uses a definition given by Mats Granvik. It adds the column T(n,0) = mu(n) at the left hand side of the triangle.

The value T(0,0) was set to -1 to make the triangle invertible as a matrix with uniform signs of the entries of the inverse.

Examples

			[ 0]                    -1,
[ 1]                   1, -1,
[ 2]                 -1, 2, -1,
[ 3]                -1, 1, 1, -1,
[ 4]              0, -1, 1, 1, -1,
[ 5]             -1, 1, 0, 0, 1, -1,
[ 6]           1, -2, 0, 1, 0, 1, -1,
[ 7]          -1, 1, 0, 0, 0, 0, 1, -1,
[ 8]        0, 0, 0, -1, 1, 0, 0, 1, -1,
[ 9]       0, 0, -1, 1, 0, 0, 0, 0, 1, -1,
The inverse of this triangle as a matrix begins
[-1,  0,  0,  0,  0,  0,  0]
[-1, -1,  0,  0,  0,  0,  0]
[-1, -2, -1,  0,  0,  0,  0]
[-1, -3, -1, -1,  0,  0,  0]
[-1, -4, -2, -1, -1,  0,  0]
[-1, -5, -2, -1, -1, -1,  0]
[-1, -6, -3, -2, -1, -1, -1]
		

Crossrefs

Cf. A123706, A008683.

Programs

Maple

with(numtheory): A206706 := proc(n,k) local moebius;
moebius := (n, k) -> `if`(k<>0 and irem(n,k) = 0, mobius(iquo(n,k)), 0);
moebius(n, k+1) - moebius(n, k) end:

Mathematica

mu[n_, k_] := If[k != 0 && Divisible[n, k], MoebiusMu[n/k], 0];
T[0, 0] = -1; T[n_, k_] /; 0 <= k <= n := mu[n, k+1] - mu[n, k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

Sage

def mur(n,k): return moebius(n//k) if k != 0 and n%k == 0 else 0
def A206706(n,k) : return -1 if n==0 and k==0 else mur(n,k+1) - mur(n,k)