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A127569 Triangle read by rows: product of the Mobius matrix A054525 by the diagonal matrix diag(A000203(n)).

%I A127569 #10 Aug 12 2015 20:49:13
%S A127569 1,-1,3,-1,0,4,0,-3,0,7,-1,0,0,0,6,1,-3,-4,0,0,12,-1,0,0,0,0,0,8,0,0,
%T A127569 0,-7,0,0,0,15,0,0,-4,0,0,0,0,0,13,1,-3,0,0,-6,0,0,0,0,18,-1,0,0,0,0,
%U A127569 0,0,0,0,0,12,0,3,0,-7,0,-12,0,0,0,0,0,28,-1,0,0,0,0,0,0,0,0,0,0,0,14,1,-3,0,0,0,0,-8,0,0,0,0,0,0,24,1,0,-4,0,-6,0,0,0,0,0
%N A127569 Triangle read by rows: product of the Mobius matrix A054525 by the diagonal matrix diag(A000203(n)).
%C A127569 Left column = mu(n), A008683; right border = sigma(n), A000203; row sums = n.
%C A127569 The definition of Mobius transform is extended to matrices here in the sense of "left multiplication by the Mobius matrix A054525". - _R. J. Mathar_, Oct 02 2007
%F A127569 T(n,k)=A054525(n,k)*A000203(k). - _R. J. Mathar_, Oct 02 2007
%e A127569 First few rows of the triangle are:
%e A127569 1;
%e A127569 -1, 3;
%e A127569 -1, 0, 4;
%e A127569 0, -3, 0, 7;
%e A127569 -1, 0, 0, 0, 6;
%e A127569 1, -3, -4, 0, 0, 12;
%e A127569 ...
%p A127569 A000203T := proc(n,k) if n = k then numtheory[sigma](n) ; else 0 ; fi ; end: A054525 := proc(n,k) if n < 1 or k > n or n mod k <> 0 then 0; else numtheory[mobius](n/k) ; fi ; end: A127569 := proc(n,k) add(A054525(n,i)*A000203T(i,k),i=1..n) ; end: for n from 1 to 15 do for k from 1 to n do printf("%a, ",A127569(n,k)) ; od: od: # _R. J. Mathar_, Oct 02 2007
%Y A127569 Cf. A054525, A000203, A008683.
%K A127569 tabl,sign
%O A127569 1,3
%A A127569 _Gary W. Adamson_, Jan 19 2007
%E A127569 Missing comma corrected by Naruto Canada, Aug 26 2007
%E A127569 More terms from _R. J. Mathar_, Oct 02 2007