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%I #6 Apr 02 2019 05:52:44
%S 1,1,2,7,14,36,90,213,520,1271,3082,7493,18238,44324,107782,262142,
%T 637368,1549870,3768886,9164499,22285034,54190024,131771616,320424614,
%U 779166270,1894671121,4607207304,11203190618,27242414612,66244451632,161084380040,391703392954
%N Expansion of 1/(1 - Sum_{k>=1} k*x^k/(1 + x^k)).
%C Invert transform of A000593.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k))).
%F G.f.: 24/(25 - theta_2(x)^4 - theta_3(x)^4), where theta_() is the Jacobi theta function.
%F a(0) = 1; a(n) = Sum_{k=1..n} A000593(k)*a(n-k).
%p a:=series(1/(1-add(k*x^k/(1+x^k),k=1..100)),x=0,32): seq(coeff(a,x,n),n=0..31); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 31; CoefficientList[Series[1/(1 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t nmax = 31; CoefficientList[Series[24/(25 - EllipticTheta[2, 0, x]^4 - EllipticTheta[3, 0, x]^4), {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 31}]
%Y Cf. A000009, A000593, A180305, A320650.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 18 2018