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%I #23 Mar 07 2020 09:04:02
%S -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,
%T -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,
%U -1,1,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,1,-1,0
%N 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.
%C 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.
%C 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.
%e [ 0] -1,
%e [ 1] 1, -1,
%e [ 2] -1, 2, -1,
%e [ 3] -1, 1, 1, -1,
%e [ 4] 0, -1, 1, 1, -1,
%e [ 5] -1, 1, 0, 0, 1, -1,
%e [ 6] 1, -2, 0, 1, 0, 1, -1,
%e [ 7] -1, 1, 0, 0, 0, 0, 1, -1,
%e [ 8] 0, 0, 0, -1, 1, 0, 0, 1, -1,
%e [ 9] 0, 0, -1, 1, 0, 0, 0, 0, 1, -1,
%e The inverse of this triangle as a matrix begins
%e [-1, 0, 0, 0, 0, 0, 0]
%e [-1, -1, 0, 0, 0, 0, 0]
%e [-1, -2, -1, 0, 0, 0, 0]
%e [-1, -3, -1, -1, 0, 0, 0]
%e [-1, -4, -2, -1, -1, 0, 0]
%e [-1, -5, -2, -1, -1, -1, 0]
%e [-1, -6, -3, -2, -1, -1, -1]
%p with(numtheory): A206706 := proc(n,k) local moebius;
%p moebius := (n, k) -> `if`(k<>0 and irem(n,k) = 0, mobius(iquo(n,k)), 0);
%p moebius(n, k+1) - moebius(n, k) end:
%t mu[n_, k_] := If[k != 0 && Divisible[n, k], MoebiusMu[n/k], 0];
%t T[0, 0] = -1; T[n_, k_] /; 0 <= k <= n := mu[n, k+1] - mu[n, k];
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)
%o (Sage)
%o def mur(n,k): return moebius(n//k) if k != 0 and n%k == 0 else 0
%o def A206706(n,k) : return -1 if n==0 and k==0 else mur(n,k+1) - mur(n,k)
%Y Cf. A123706, A008683.
%K sign,tabl
%O 0,5
%A _Peter Luschny_, Feb 11 2012
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