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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
 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, -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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A051731 = the inverse of this triangle = A129372 * A115361. - Gary W. Adamson, Apr 15 2007 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 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows Trevor Hyde, Cyclotomic factors of necklace polynomials, arXiv:1811.08601 [math.CO], 2018. N. Metropolis, G.-C. Rota, Witt vectors and the algebra of necklaces, Adv. Math. 50 (1983), 95-125. Pieter Moree, The formal series Witt transform, Discr. Math. 295 (2005), 143-160. FORMULA Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006 Equals = A129360 * A115359 as infinite lower triangular matrices. - Gary W. Adamson, Apr 15 2007 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 EXAMPLE Triangle (with rows n >= 1 and columns k >= 1) begins as follows:    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, -1,  0,  0,  0,  1; ... Matrix inverse is triangle A051731:   1;   1, 1;   1, 0, 1;   1, 1, 0, 1;   1, 0, 0, 0, 1;   1, 1, 1, 0, 0, 1;   1, 0, 0, 0, 0, 0, 1;   1, 1, 0, 1, 0, 0, 0, 1; ... MAPLE A054525 := proc(n, k)     if n mod k = 0 then         numtheory[mobius](n/k) ;     else         0 ;     end if; end proc: # R. J. Mathar, Oct 21 2012 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A054521. Cf. A051731, A115361, A129372. Cf. A077050, A115359, A129360. Sequence in context: A115524 A117198 A271047 * A174852 A065333 A244611 Adjacent sequences:  A054522 A054523 A054524 * A054526 A054527 A054528 KEYWORD sign,tabl AUTHOR N. J. A. Sloane, Apr 09 2000 STATUS approved

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Last modified October 17 14:26 EDT 2019. Contains 328113 sequences. (Running on oeis4.)