cp's OEIS Frontend

A118206 Euler transform of the Liouville function.

%I A118206 #31 Sep 02 2023 20:31:19
%S A118206 1,1,0,-1,0,0,0,-1,-1,0,2,0,-2,-2,1,2,2,-2,-2,0,2,-1,-1,-2,2,5,4,-5,
%T A118206 -5,-2,4,2,-2,-7,3,8,5,-7,-6,1,14,4,-9,-14,2,5,5,-10,-7,6,22,3,-12,
%U A118206 -20,1,15,15,-16,-12,4,25,6,-14,-31,13,33,14,-39,-32,-6,39,15,-20,-31,33,41,14,-53,-44,3,66,12,-35,-51,22,48,36,-60,-43,21
%N A118206 Euler transform of the Liouville function.
%H A118206 Paul D. Hanna, <a href="/A118206/b118206.txt">Table of n, a(n) for n = 0..1000</a>
%H A118206 Peter Bala, <a href="/A118205/a118205.pdf">Cyclotomic polynomials and the generating functions of A118205 and A118206</a>
%H A118206 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>
%H A118206 Wikipedia, <a href="https://en.wikipedia.org/wiki/Liouville_function">Liouville function</a>
%F A118206 G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
%F A118206 Logarithmic derivative yields A061020. - _Paul D. Hanna_, Sep 22 2011
%F A118206 G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - _Peter Bala_, Mar 31 2023
%t A118206 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 A118206 max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 23 2015 *)
%o A118206 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*moebius(core(d)))*x^m/m)+x*O(x^n)),n)} /* Cf. A061020 - _Paul D. Hanna_, Sep 22 2011 */
%Y A118206 Cf. A118205, A118207, A118208, A061020, A117209.
%K A118206 sign,easy
%O A118206 0,11
%A A118206 _Stuart Clary_, Apr 15 2006