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A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

%I A317529 #22 Dec 18 2023 01:42:56
%S A317529 1,-1,1,0,1,-1,1,-2,2,-1,1,0,1,-1,1,-1,1,-2,1,0,1,-1,1,-2,2,-1,2,0,1,
%T A317529 -1,1,-3,1,-1,1,0,1,-1,1,-2,1,-1,1,0,2,-1,1,-1,2,-2,1,0,1,-2,1,-2,1,
%U A317529 -1,1,0,1,-1,2,-2,1,-1,1,0,1,-1,1,-4,1,-1,2,0,1,-1,1,-1,3,-1,1,0,1,-1,1,-2,1,-2,1,0,1,-1,1
%N A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).
%H A317529 Antti Karttunen, <a href="/A317529/b317529.txt">Table of n, a(n) for n = 1..11025</a>
%H A317529 Antti Karttunen, <a href="/A317529/a317529.txt">Data supplement: n, a(n) computed for n =  1..100000</a>
%F A317529 G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)).
%F A317529 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
%F A317529 a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d).
%F A317529 If n is odd, a(n) = A046951(n).
%F A317529 Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - _Amiram Eldar_, Oct 25 2020
%F A317529 From _Amiram Eldar_, Dec 18 2023: (Start)
%F A317529 Dirichlet g.f.: zeta(s) * zeta(2*s) * (1 - 1/2^(s-1)).
%F A317529 Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = -(sqrt(2)-1) * zeta(1/2) = 0.604898... (A113024). (End)
%p A317529 seq(coeff(series(add(x^(k^2)/(1+x^(k^2)),k=1..n), x,n+1),x,n),n=1..100); # _Muniru A Asiru_, Jul 30 2018
%t A317529 nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A317529 nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t A317529 Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}]
%t A317529 f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o A317529 (PARI) A317529(n) = sumdiv(n,d,((-1)^(1+(n/d)))*issquare(d)); \\ _Antti Karttunen_, Nov 07 2018
%Y A317529 Cf. A000290, A010052, A046951, A048272, A113024, A300853.
%K A317529 sign,mult,easy
%O A317529 1,8
%A A317529 _Ilya Gutkovskiy_, Jul 30 2018