OFFSET
0,5
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.
EXAMPLE
[ 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]
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 *)
PROG
(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)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Feb 11 2012
STATUS
approved