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A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).

%I A328640 #12 Dec 03 2022 05:44:37
%S A328640 1,-9,-28,9,-126,252,-344,-9,28,1134,-1332,-252,-2198,3096,3528,9,
%T A328640 -4914,-252,-6860,-1134,9632,11988,-12168,252,126,19782,-28,-3096,
%U A328640 -24390,-31752,-29792,-9,37296,44226,43344,252,-50654,61740,61544,1134,-68922,-86688,-79508,-11988,-3528
%N A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).
%C A328640 Dirichlet inverse of A065959.
%H A328640 Amiram Eldar, <a href="/A328640/b328640.txt">Table of n, a(n) for n = 1..10000</a>
%F A328640 a(1) = 1; a(n) = -Sum_{d|n, d<n} A065959(n/d) * a(d).
%F A328640 a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.
%F A328640 Multiplicative with a(p^) = (-1)^e*(p^3+1). - _Amiram Eldar_, Dec 03 2022
%t A328640 a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
%t A328640 Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
%t A328640 f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 03 2022 *)
%o A328640 (PARI) a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ _Andrew Howroyd_, Oct 25 2019
%Y A328640 Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.
%K A328640 sign,mult
%O A328640 1,2
%A A328640 _Ilya Gutkovskiy_, Oct 23 2019