A338549 a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

1, -17, 80, -240, 624, -1360, 2400, -3840, 6480, -10608, 14640, -19200, 28560, -40800, 49920, -61440, 83520, -110160, 130320, -149760, 192000, -248880, 279840, -307200, 390000, -485520, 524880, -576000, 707280, -848640, 923520, -983040, 1171200, -1419840, 1497600, -1555200

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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).

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

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).

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

Mathematica

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

Prog

(PARI) a(n) = n^4 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^4); \\ Michel Marcus, Nov 02 2020

Crossrefs

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

Sequence in context: A142021 A353937 A172045 * A060934 A228602 A100688

Adjacent sequences: A338546 A338547 A338548 * A338550 A338551 A338552

Keyword

sign,mult

Author

Ilya Gutkovskiy, Nov 02 2020

Status

approved