A118205 - OEIS

%I #23 Sep 02 2023 20:32:20

%S 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,

%T -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,

%U -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

%N Euler transform of the negative of the Liouville function.

%H Peter Bala, <a href="/A118205/a118205.pdf">Cyclotomic polynomials and the generating functions of A118205 and A118206</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Liouville_function">Liouville function</a>

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

%F 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

%t 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 ]

%t (* Second program (needs Mma >= 7.0): *)

%t nmax = 100;

%t Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* _Jean-François Alcover_, Jan 08 2020 *)

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

%K sign,easy

%O 0,6

%A _Stuart Clary_, Apr 15 2006