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%I #29 Sep 02 2023 20:31:35
%S 1,1,-1,-2,1,2,0,-2,-2,0,5,2,-7,-6,7,9,0,-10,-9,4,17,2,-18,-12,14,21,
%T 5,-26,-25,14,41,4,-38,-35,18,53,23,-56,-54,31,86,15,-78,-85,34,112,
%U 41,-110,-102,49,158,40,-138,-150,68,195,68,-191,-190,69,279,89,-217,-253,102,327,122,-336,-335,118,462,142,-361,-430,170
%N Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.
%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 From _Peter Bala_, Apr 05 2023: (Start)
%F G.f.: A(x) = Product_{k >= 1} C(k,x^(2*k)) / C(k,x^k) = Product_{k >= 1} C(2*k,x^k) / C(4*k,x^k) = -Product_{k >= 1} C(k,x^(2*k)) * C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial.
%F Conjecture: A(x^2) = Product_{k >= 1} C(k,x^k) * C(k,(-x)^k). (End)
%t nmax = 80; 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 (* From version 7 on *) nmax = 80; CoefficientList[ Series[ Product[ (1 + x^k)^LiouvilleLambda[k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Jean-François Alcover_, Jul 30 2013 *)
%Y Cf. A117210, A118205, A118206, A118208, A118209.
%K sign,easy
%O 0,4
%A _Stuart Clary_, Apr 15 2006
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