A118205 Euler transform of the negative of the Liouville function.

1, -1, 1, 0, -1, 2, -2, 2, 0, -2, 3, -2, 1, 2, -3, 3, -2, 0, 3, -2, 3, -2, 0, 2, -2, 3, -1, 0, 1, -2, 5, 0, 0, 1, -2, 1, 1, 2, 0, 1, -2, 1, 4, -1, 4, -2, -3, 6, -2, 5, 6, -8, 6, -4, 2, 9, -8, 7, -4, -1, 11, -1, 5, 1, -8, 5, 2, 4, 7, -8, 4, 2, 1, 14, -2, 0, -1, -6, 19, 2, 5, 6, -15, 12, 1, 3, 18, -17, 1, 9, 0, 29, -4, -3, 4, -13, 14, 17, 2, 0, -4

Offset

0,6

Links

Table of n, a(n) for n=0..100.

Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206

Wikipedia, Cyclotomic polynomial

Wikipedia, Liouville function

Formula

G.f.: A(x) = Product_{k>=1} (1 - x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

G.f.: A(x) = - Product_{k >= 1} C(k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

Mathematica

nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* Second program (needs Mma >= 7.0): *)
nmax = 100;
Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Jan 08 2020 *)

Crossrefs

Cf. A061020, A117208, A118206, A118207, A118208.

Sequence in context: A118664 A223175 A322213 * A216265 A336694 A130277

Adjacent sequences: A118202 A118203 A118204 * A118206 A118207 A118208

Keyword

sign,easy

Author

Stuart Clary, Apr 15 2006

Status

approved