%I #21 Sep 08 2022 08:46:23
%S 1,-2,-4,4,8,16,-12,-28,-28,-56,64,68,152,144,-20,-72,-678,-508,-424,
%T 92,824,1512,2204,1036,936,-1900,-2936,-6444,-5656,-4384,-4808,6540,
%U 10080,21256,20296,24424,13520,-7856,-28836,-55744,-72240,-92960,-48424,-40772,36168,106464,199996
%N Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))/(1 + x^(i*j*k)).
%C Convolution inverse of A305050.
%H Seiichi Manyama, <a href="/A321241/b321241.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^A007425(k).
%F G.f.: Product_{k>=1} theta_4(x^k)^tau(k), where theta_4() is the Jacobi theta function and tau() is the number of divisors. - _Ilya Gutkovskiy_, May 18 2019
%t With[{nmax=50}, CoefficientList[Series[Product[(1 - x^(i*j*k))/(1 + x^(i*j*k)), {i,1,nmax}, {j,1,nmax/i}, {k,1,nmax/i/j}], {x,0,nmax}], x]] (* _G. C. Greubel_, Nov 01 2018 *)
%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1,m, ((1-x^k)/(1+x^k))^sumdiv(k, x, sumdiv(x, y, 1 )))) \\ _G. C. Greubel_, Nov 01 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(&*[(1 - x^(i*j*k))/(1 + x^(i*j*k)): i in [1..m]]): j in [1..m]]): k in [1..m]]))); // _G. C. Greubel_, Nov 01 2018
%Y Cf. A000005, A002448, A007425, A305050, A320908.
%K sign
%O 0,2
%A _Seiichi Manyama_, Nov 01 2018