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A328372 Expansion of Sum_{k>=1} x^(k^2) / (1 - x^(2*k^2)).

%I A328372 #19 Sep 08 2022 08:46:24
%S A328372 1,0,1,1,1,0,1,0,2,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,2,0,2,1,1,0,1,0,1,0,
%T A328372 1,2,1,0,1,0,1,0,1,1,2,0,1,1,2,0,1,1,1,0,1,0,1,0,1,1,1,0,2,1,1,0,1,1,
%U A328372 1,0,1,0,1,0,2,1,1,0,1,1,3,0,1,1,1,0,1,0,1,0,1
%N A328372 Expansion of Sum_{k>=1} x^(k^2) / (1 - x^(2*k^2)).
%C A328372 Number of odd divisors d of n such that n/d is square.
%C A328372 Inverse Moebius transform of A143259.
%H A328372 Amiram Eldar, <a href="/A328372/b328372.txt">Table of n, a(n) for n = 1..10000</a>
%F A328372 G.f.: Sum_{k>=1} (theta_3(x^(2*k - 1)) - 1) / 2.
%F A328372 Dirichlet g.f.: (1 - 1/2^s) * zeta(s) * zeta(2*s).
%F A328372 a(n) = Sum_{d|n} A000035(d) * A010052(n/d).
%F A328372 Sum_{k=1..n} a(k) ~ Pi^2 * n / 12 + (sqrt(2)-1) * zeta(1/2) * sqrt(n/2). - _Vaclav Kotesovec_, Oct 14 2019
%F A328372 Multiplicative with a(2^e) = 0 if e is odd, and 1 if e is even, and for p > 2, a(p^e) = ceiling((e+1)/2). - _Amiram Eldar_, Oct 16 2020
%t A328372 nmax = 91; CoefficientList[Series[Sum[x^(k^2)/(1 -  x^(2 k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t A328372 Table[DivisorSum[n, 1 &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 91}]
%t A328372 Table[DivisorSum[n, (-1)^PrimeOmega[#] Mod[DivisorSigma[1, #], 2] &], {n, 1, 91}]
%t A328372 f[p_, e_] := If[p == 2, Boole @ EvenQ[e], Ceiling[(e + 1)/2]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 16 2020 *)
%o A328372 (PARI) a(n) = sumdiv(n, d, (d%2) && issquare(n/d)); \\ _Michel Marcus_, Oct 14 2019
%o A328372 (Magma) [#[d:d in Divisors(n)| IsOdd(d) and IsSquare(n div d)]:n in [1..90]]; // _Marius A. Burtea_, Oct 14 2019
%Y A328372 Cf. A000035, A001227, A010052, A036554 (positions of 0's), A046951, A143259, A328373.
%K A328372 nonn,mult,easy
%O A328372 1,9
%A A328372 _Ilya Gutkovskiy_, Oct 14 2019