%I A054525
%S A054525 1,1,1,1,0,1,0,1,0,1,1,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0,0,0,
%T A054525 1,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,
%U A054525 0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1
%V A054525 1,-1,1,-1,0,1,0,-1,0,1,-1,0,0,0,1,1,-1,-1,0,0,1,-1,0,0,0,0,0,1,0,0,0,
%W A054525 -1,0,0,0,1,0,0,-1,0,0,0,0,0,1,1,-1,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,
%X A054525 0,0,0,1,0,1,0,-1,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,-1
%N A054525 Triangle T(n,k): T(n,k) = mu(n/k) if k divides n, T(n,k)=0 otherwise 
               (n >= 1, 1<=k<=n).
%C A054525 A051731 = the inverse of this triangle = A129372 * A115361. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%C A054525 If a column T(n,0)=0 is added, these are the coefficients of the necklace 
               polynomials multiplied by n [Moree,Metropolis]. [From R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Nov 11 2008]
%H A054525 Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">
               The formal series Witt transform</a>, Discr. Math. 295 (2005), 143-160. 
               [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 2008]
%H A054525 N. Metropolis, G.-C. Rota, <a href="http://dx.doi.org/10.1016/0001-8708(83)90035-X">
               Witt vectors and the algebra of necklaces</a>, Adv. Math. 50 (1983), 
               95-125. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 
               2008]
%F A054525 Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 
               otherwise. - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 09 2006
%F A054525 Equals = A129360 * A115369 as infinite lower triangular matrices. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007
%e A054525 Triangle begins:
%e A054525 1;
%e A054525 -1,1;
%e A054525 -1,0,1;
%e A054525 0,-1,0,1;
%e A054525 -1,0,0,0,1;
%e A054525 1,-1,-1,0,0,1;
%e A054525 -1,0,0,0,0,0,1;
%e A054525 0,0,0,-1,0,0,0,1; ...
%e A054525 Matrix inverse is triangle A051731:
%e A054525 1;
%e A054525 1,1;
%e A054525 1,0,1;
%e A054525 1,1,0,1;
%e A054525 1,0,0,0,1;
%e A054525 1,1,1,0,0,1;
%e A054525 1,0,0,0,0,0,1;
%e A054525 1,1,0,1,0,0,0,1; ...
%Y A054525 Cf. A054521.
%Y A054525 Cf. A051731.
%Y A054525 Cf. A129360, A115369.
%Y A054525 Sequence in context: A115952 A115524 A117198 this_sequence A065333 A127972 
               A103451
%Y A054525 Adjacent sequences: A054522 A054523 A054524 this_sequence A054526 A054527 
               A054528
%K A054525 sign,tabl
%O A054525 1,1
%A A054525 N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2000