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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). 95
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

Pieter Moree, The formal series Witt transform, Discr. Math. 295 (2005), 143-160.

N. Metropolis, G.-C. Rota, Witt vectors and the algebra of necklaces, Adv. Math. 50 (1983), 95-125.

FORMULA

Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006

Equals = A129360 * A115369 as infinite lower triangular matrices. - Gary W. Adamson, Apr 15 2007

EXAMPLE

Triangle begins:

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-Fran├žois 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.

Cf. A129360, A115369.

Sequence in context: A115952 A115524 A117198 * 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 September 24 21:29 EDT 2017. Contains 292441 sequences.