%I #11 Nov 16 2022 06:55:36

%S 1,-17,80,-240,624,-1360,2400,-3840,6480,-10608,14640,-19200,28560,

%T -40800,49920,-61440,83520,-110160,130320,-149760,192000,-248880,

%U 279840,-307200,390000,-485520,524880,-576000,707280,-848640,923520,-983040,1171200,-1419840,1497600,-1555200

%N a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

%H Amiram Eldar, <a href="/A338549/b338549.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 11*x^k + 11*x^(2*k) - x^(3*k)) / (1 + x^k)^5.

%F G.f. A(x) satisfies: A(x) = x * (1 - 11*x + 11*x^2 - x^3) / (1 + x)^5 - Sum_{k>=2} A(x^k).

%F Dirichlet g.f.: (1 - 2^(5 - s)) * zeta(s - 4) / zeta(s).

%F a(n) = J_4(n) if n odd, J_4(n) - 32 * J_4(n/2) if n even, where J_4 = A059377 (Jordan function J_4).

%F Multiplicative with a(2) = -17, a(2^e) = -15*2^(4*(e-1)) for e > 1, and a(p^e) = (p^4-1)*p^(4*(e-1)) for p > 2. - _Amiram Eldar_, Nov 15 2022

%t Table[n^4 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^4, {d, Divisors[n]}], {n, 1, 36}]

%t nmax = 36; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 11 x^k + 11 x^(2 k) - x^(3 k))/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t f[p_, e_] := (p^4 - 1)*p^(4*(e - 1)); f[2, 1] = -17; f[2, e_] := -15*2^(4*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2022 *)

%o (PARI) a(n) = n^4 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^4); \\ _Michel Marcus_, Nov 02 2020

%Y Cf. A008683, A059377, A279395, A325596, A338547, A338548.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Nov 02 2020