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%I #48 Aug 24 2021 06:47:05
%S 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 -1,0,0,0,1,0,0,-1,0,0,0,0,0,1,1,-1,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,
%U 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
%N 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 A051731 = the inverse of this triangle = A129372 * A115361. - _Gary W. Adamson_, Apr 15 2007
%C If a column T(n,0)=0 is added, these are the coefficients of the necklace polynomials multiplied by n [Moree, Metropolis]. - _R. J. Mathar_, Nov 11 2008
%H G. C. Greubel, <a href="/A054525/b054525.txt">Table of n, a(n) for the first 50 rows</a>
%H Trevor Hyde, <a href="https://arxiv.org/abs/1811.08601">Cyclotomic factors of necklace polynomials</a>, arXiv:1811.08601 [math.CO], 2018.
%H N. Metropolis and 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.
%H 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.
%F Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 otherwise. - _Paul D. Hanna_, Jan 09 2006
%F Equals = A129360 * A115359 as infinite lower triangular matrices. - _Gary W. Adamson_, Apr 15 2007
%F Bivariate g.f.: Sum_{n, k >= 1} T(n, k)*x^n*y^k = Sum_{m >= 1} mu(m)*x^m*y/(1 - x^m*y). - _Petros Hadjicostas_, Jun 25 2019
%e Triangle (with rows n >= 1 and columns k >= 1) begins as follows:
%e 1;
%e -1, 1;
%e -1, 0, 1;
%e 0, -1, 0, 1;
%e -1, 0, 0, 0, 1;
%e 1, -1, -1, 0, 0, 1;
%e -1, 0, 0, 0, 0, 0, 1;
%e 0, 0, 0, -1, 0, 0, 0, 1; ...
%e Matrix inverse is triangle A051731:
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, 1, 0, 1;
%e 1, 0, 0, 0, 1;
%e 1, 1, 1, 0, 0, 1;
%e 1, 0, 0, 0, 0, 0, 1;
%e 1, 1, 0, 1, 0, 0, 0, 1; ...
%p A054525 := proc(n,k)
%p if n mod k = 0 then
%p numtheory[mobius](n/k) ;
%p else
%p 0 ;
%p end if;
%p end proc: # _R. J. Mathar_, Oct 21 2012
%t t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k ], 0]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%o (PARI) tabl(nn) = {T = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); for (n=1, nn, for (k=1, n, print1(T[n, k], ", ");); print(););} \\ _Michel Marcus_, Mar 28 2015
%o (PARI) row(n) = Vecrev(sumdiv(n, d, moebius(d)*x^(n/d))/x); \\ _Michel Marcus_, Aug 24 2021
%Y Cf. A054521.
%Y Cf. A051731, A115361, A129372.
%Y Cf. A077050, A115359, A129360.
%K sign,tabl
%O 1,1
%A _N. J. A. Sloane_, Apr 09 2000
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