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 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). 96

%I

%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, 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

%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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Last modified September 18 09:54 EDT 2019. Contains 327168 sequences. (Running on oeis4.)