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A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

%I A327096 #15 Oct 23 2022 02:53:46
%S A327096 1,3,5,7,7,15,9,15,18,21,13,35,15,27,35,31,19,54,21,49,45,39,25,75,38,
%T A327096 45,58,63,31,105,33,63,65,57,63,126,39,63,75,105,43,135,45,91,126,75,
%U A327096 49,155,66,114,95,105,55,174,91,135,105,93,61,245,63,99
%N A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.
%C A327096 Inverse Moebius transform of A002131.
%C A327096 Dirichlet convolution of A000027 with A001227.
%H A327096 Antti Karttunen, <a href="/A327096/b327096.txt">Table of n, a(n) for n = 1..20000</a>
%F A327096 G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
%F A327096 G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
%F A327096 a(n) = Sum_{d|n} A002131(d).
%F A327096 a(n) = Sum_{d|n} d * A001227(n/d).
%F A327096 a(n) = (A007429(n) + A288417(n)) / 2.
%F A327096 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - _Amiram Eldar_, Oct 23 2022
%t A327096 nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A327096 a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]
%o A327096 (PARI) a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ _Andrew Howroyd_, Sep 13 2019
%Y A327096 Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.
%K A327096 nonn,mult
%O A327096 1,2
%A A327096 _Ilya Gutkovskiy_, Sep 13 2019