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A206706
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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.
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2
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-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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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EXAMPLE
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[ 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]
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MAPLE
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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:
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MATHEMATICA
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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];
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PROG
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(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)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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