A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.

1, 0, 5, -5, 7, 0, 9, -18, 18, 0, 13, -25, 15, 0, 35, -47, 19, 0, 21, -35, 45, 0, 25, -90, 38, 0, 58, -45, 31, 0, 33, -108, 65, 0, 63, -90, 39, 0, 75, -126, 43, 0, 45, -65, 126, 0, 49, -235, 66, 0, 95, -75, 55, 0, 91, -162, 105, 0, 61, -175, 63, 0, 162, -233, 105

Offset

1,3

Comments

Inverse Moebius transform of A002129.

Dirichlet convolution of A000005 with A181983.

Links

Table of n, a(n) for n=1..65.

Formula

a(n) = Sum_{d|n} A002129(d).

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

Mathematica

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[(-1)^(d + 1) d DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]

Prog

(PARI) a(n) = {sumdiv(n, d, (-1)^(d + 1) * d * numdiv(n/d))} \\ Andrew Howroyd, Sep 14 2019
(Magma) [&+[(-1)^(d+1)*d*#Divisors(n div d):d in Divisors(n)]:n in [1..65]]; // Marius A. Burtea, Sep 14 2019

Crossrefs

Cf. A000005, A002129, A007429, A008586 (positions of negative terms), A016825 (positions of 0's), A181983, A288417, A288571.

Sequence in context: A008945 A008705 A348293 * A173932 A249649 A226571

Adjacent sequences: A327239 A327240 A327241 * A327243 A327244 A327245

Keyword

sign,mult

Author

Ilya Gutkovskiy, Sep 14 2019

Status

approved