A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

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, 5, -26, -25, 14, 41, 4, -38, -35, 18, 53, 23, -56, -54, 31, 86, 15, -78, -85, 34, 112, 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

Offset

0,4

Links

Table of n, a(n) for n=0..74.

Wikipedia, Cyclotomic polynomial

Wikipedia, Liouville function

Formula

From Peter Bala, Apr 05 2023: (Start)

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.

Conjecture: A(x^2) = Product_{k >= 1} C(k,x^k) * C(k,(-x)^k). (End)

Mathematica

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 ]
(* 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 *)

Crossrefs

Cf. A117210, A118205, A118206, A118208, A118209.

Sequence in context: A112181 A151676 A151684 * A327274 A055378 A217921

Adjacent sequences: A118204 A118205 A118206 * A118208 A118209 A118210

Keyword

sign,easy

Author

Stuart Clary, Apr 15 2006

Status

approved