%I #13 Nov 16 2022 06:55:49

%S 1,-5,8,-12,24,-40,48,-48,72,-120,120,-96,168,-240,192,-192,288,-360,

%T 360,-288,384,-600,528,-384,600,-840,648,-576,840,-960,960,-768,960,

%U -1440,1152,-864,1368,-1800,1344,-1152,1680,-1920,1848,-1440,1728,-2640,2208,-1536,2352,-3000

%N a(n) = n^2 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^2.

%C Moebius transform of A162395.

%H Amiram Eldar, <a href="/A338547/b338547.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} mu(k) * x^k * (1 - x^k) / (1 + x^k)^3.

%F G.f. A(x) satisfies: A(x) = x * (1 - x) / (1 + x)^3 - Sum_{k>=2} A(x^k).

%F Dirichlet g.f.: (1 - 2^(3 - s)) * zeta(s - 2) / zeta(s).

%F a(n) = J_2(n) if n odd, J_2(n) - 8 * J_2(n/2) if n even, where J_2 = A007434 (Jordan function J_2).

%F Multiplicative with a(2) = -5, a(2^e) = -3*2^(2*(e-1)) for e > 1, and a(p^e) = (p^2-1)*p^(2*(e-1)) for p > 2. - _Amiram Eldar_, Nov 15 2022

%t Table[n^2 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^2, {d, Divisors[n]}], {n, 1, 50}]

%t nmax = 50; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t f[p_, e_] := (p^2 - 1)*p^(2*(e - 1)); f[2, 1] = -5; f[2, e_] := -3*2^(2*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2022 *)

%o (PARI) a(n) = n^2 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^2); \\ _Michel Marcus_, Nov 02 2020

%Y Cf. A007434, A008683, A162395, A321543, A325596, A338548, A338549.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Nov 02 2020