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A123706 Matrix inverse of triangle A010766, where A010766(n,k) = [n/k], for n>=k>=1.

%I A123706 #22 Apr 23 2017 01:03:12
%S A123706 1,-2,1,-1,-1,1,1,-1,-1,1,-1,0,0,-1,1,2,0,-1,0,-1,1,-1,0,0,0,0,-1,1,0,
%T A123706 0,1,-1,0,0,-1,1,0,1,-1,0,0,0,0,-1,1,2,-1,0,1,-1,0,0,0,-1,1,-1,0,0,0,
%U A123706 0,0,0,0,0,-1,1,-1,1,1,-1,1,-1,0,0,0,0,-1,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,2,-1,0,0,0,1,-1,0,0,0,0,0,-1,1,1,1,-1,1,-1,0,0,0
%N A123706 Matrix inverse of triangle A010766, where A010766(n,k) = [n/k], for n>=k>=1.
%C A123706 Unsigned elements consist of only 0's, 1's and 2's.
%H A123706 Enrique Pérez Herrero, <a href="/A123706/b123706.txt">Rows n = 1..100 of triangle, flattened</a>
%F A123706 T(n,1) = +2 when n = 2*p where p is an odd prime.
%F A123706 T(n,1) = -2 when n is an even squarefree number with an odd number of prime divisors.
%F A123706 A123709(n) = number of nonzero terms in row n = 2^(m+1) - 1 when n is an odd number with exactly m distinct prime factors.
%F A123706 Sum_{k=1..n} T(n,k) = moebius(n).
%F A123706 Sum_{k=1..n} T(n,k)*k = 0 for n>1.
%F A123706 Sum_{k=1..n} T(n,k)*k^2 = 2*phi(n) for n>1 where phi(n)=A000010(n).
%F A123706 Sum_{k=1..n} T(n,k)*k^3 = 6*A102309(n) for n>1 where A102309(n)=Sum[d|n, moebius(d)*C(n/d,2) ].
%F A123706 Sum_{k=1..n} T(n,k)*k*2^(k-1) = A085411(n) = Sum_{d|n} mu(n/d)*(d+1)*2^(d-2) = total number of parts in all compositions of n into relatively prime parts.
%F A123706 T(n,k) = mu(n/k)-mu(n/(k+1)), where mu(n/k) is A008683(n/k) if k|n and 0 otherwise. - _Enrique Pérez Herrero_, Feb 21 2012
%e A123706 Triangle begins:
%e A123706 1;
%e A123706 -2, 1;
%e A123706 -1,-1, 1;
%e A123706 1,-1,-1, 1;
%e A123706 -1, 0, 0,-1, 1;
%e A123706 2, 0,-1, 0,-1, 1;
%e A123706 -1, 0, 0, 0, 0,-1, 1;
%e A123706 0, 0, 1,-1, 0, 0,-1, 1;
%e A123706 0, 1,-1, 0, 0, 0, 0,-1, 1;
%e A123706 2,-1, 0, 1,-1, 0, 0, 0,-1, 1;
%e A123706 -1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;
%e A123706 -1, 1, 1,-1, 1,-1, 0, 0, 0, 0,-1, 1;
%e A123706 -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1;
%e A123706 2,-1, 0, 0, 0, 1,-1, 0, 0, 0, 0, 0,-1, 1;
%e A123706 1, 1,-1, 1,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1; ...
%t A123706 t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k+1], MoebiusMu[n/(k+1)], 0]; Table[t[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 29 2013, after _Enrique Pérez Herrero_ *)
%o A123706 (PARI) T(n,k)=(matrix(n,n,r,c,r\c)^-1)[n,k]  \\ simplified by _M. F. Hasler_, Feb 12 2012
%Y A123706 Cf. A102309, A085411; A123707, A123708, A123709.
%K A123706 sign,tabl
%O A123706 1,2
%A A123706 _Paul D. Hanna_, Oct 09 2006