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%I #17 Sep 15 2023 05:44:45
%S 1,-4,-6,4,-10,24,-14,0,9,40,-22,-24,-26,56,60,0,-34,-36,-38,-40,84,
%T 88,-46,0,25,104,0,-56,-58,-240,-62,0,132,136,140,36,-74,152,156,0,
%U -82,-336,-86,-88,-90,184,-94,0,49,-100,204,-104,-106,0,220,0,228,232,-118,240
%N Dirichlet g.f.: 1 / zeta(s-1)^2.
%C Dirichlet inverse of A038040.
%C Dirichlet convolution of A055615 with itself.
%C Moebius transform of A101035.
%H Antti Karttunen, <a href="/A328722/b328722.txt">Table of n, a(n) for n = 1..20000</a>
%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A038040(n/d) * a(d).
%F a(n) = n * A007427(n).
%F a(n) = Sum_{d|n} mu(n/d) * A101035(d).
%F Multiplicative with a(p) = -2*p, a(p^2) = p^2, and a(p) = 0 for e >= 3. - _Amiram Eldar_, Sep 15 2023
%t a[1] = 1; a[n_] := -Sum[(n/d) DivisorSigma[0, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]
%t Table[n DivisorSum[n, MoebiusMu[n/#] MoebiusMu[#] &], {n, 1, 60}]
%t f[p_, e_] := Switch[e, 1, -2*p, 2, p^2, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2023 *)
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 22 2021
%o (PARI)
%o A007427(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d))); \\ From A007427
%o A328722(n) = (n*A007427(n)); \\ _Antti Karttunen_, Nov 15 2021
%Y Cf. A007427, A008683, A038040, A046099 (positions of 0's), A055615, A101035.
%K sign,easy,mult
%O 1,2
%A _Ilya Gutkovskiy_, Oct 26 2019
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