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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).

%I A054525 #52 Nov 13 2024 14:56:14
%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,-1,0,0,0,0,0,0,0,
%U A054525 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 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_, 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]. - _R. J. Mathar_, Nov 11 2008
%H A054525 G. C. Greubel, <a href="/A054525/b054525.txt">Table of n, a(n) for the first 50 rows</a>
%H A054525 Trevor Hyde, <a href="https://arxiv.org/abs/1811.08601">Cyclotomic factors of necklace polynomials</a>, arXiv:1811.08601 [math.CO], 2018.
%H A054525 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 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.
%F A054525 Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 otherwise. - _Paul D. Hanna_, Jan 09 2006
%F A054525 Equals = A129360 * A115359 as infinite lower triangular matrices. - _Gary W. Adamson_, Apr 15 2007
%F A054525 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 A054525 Triangle (with rows n >= 1 and columns k >= 1) begins as follows:
%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; ...
%p A054525 A054525 := proc(n,k)
%p A054525     if n mod k = 0 then
%p A054525         numtheory[mobius](n/k) ;
%p A054525     else
%p A054525         0 ;
%p A054525     end if;
%p A054525 end proc: # _R. J. Mathar_, Oct 21 2012
%t A054525 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 A054525 (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 A054525 (PARI) row(n) = Vecrev(sumdiv(n, d, moebius(d)*x^(n/d))/x); \\ _Michel Marcus_, Aug 24 2021
%o A054525 (Python)
%o A054525 from math import isqrt, comb
%o A054525 from sympy import mobius
%o A054525 def A054525(n): return 0 if (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))%(b:=n-comb(a,2)) else mobius(a//b) # _Chai Wah Wu_, Nov 13 2024
%Y A054525 Cf. A054521.
%Y A054525 Cf. A051731, A115361, A129372.
%Y A054525 Cf. A077050, A115359, A129360.
%K A054525 sign,tabl
%O A054525 1,1
%A A054525 _N. J. A. Sloane_, Apr 09 2000