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URL: https://oeis.org/A327242

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A327242
Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.
1
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
FORMULA
a(n) = Sum_{d|n} A002129(d).
a(n) = Sum_{d|n} (-1)^(d + 1) * d * tau(n/d).
Multiplicative with a(2^e) = 3*e + 5 - 2^(e+2), and a(p^e) = (p^(e+2) - (e+2)*p +e + 1)/(p-1)^2 for an odd prime p. - Amiram Eldar, May 25 2025
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}]
f[p_, e_] := (p^(e + 2) - (e + 2)*p + e + 1)/(p-1)^2; f[2, e_] := 3*e + 5 - 2^(e+2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)
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: A394102 A008705 A348293 * A173932 A249649 A226571
KEYWORD
sign,mult
AUTHOR
Ilya Gutkovskiy, Sep 14 2019
STATUS
approved