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%I #35 Sep 08 2022 08:46:19
%S 1,-1,-1,-1,1,1,0,2,2,-1,-2,0,-1,-1,-4,-1,2,0,-1,2,2,5,-1,4,8,-4,-5,0,
%T -1,-1,-6,-1,3,-7,-9,-5,1,3,-3,3,17,0,-6,8,12,8,0,8,17,-11,-9,-10,0,
%U -2,-20,5,14,-18,-25,-10,1,-7,-21,2,29,-12,-17,6,17,32,-4
%N Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
%H Seiichi Manyama, <a href="/A288007/b288007.txt">Table of n, a(n) for n = 0..10000</a>
%F Convolution inverse of A107742.
%F a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0.
%F G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^(2*k)))). - _Ilya Gutkovskiy_, Aug 26 2018
%p with(numtheory): seq(coeff(series(exp(-add(sigma(k)*x^k/(k*(1-x^(2*k))),k=1..n)),x,n+1), x, n), n = 0 .. 70); # _Muniru A Asiru_, Jan 30 2019
%t A109386[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2] &] &]; a[0] = 1; a[n_] := a[n] = -(1/n) Sum[A109386[k] a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jun 04 2017 *)
%t CoefficientList[Series[1/Product[Product[1+x^(j*k), {j,1,100}], {k,1,100}], {x,0,80}], x] (* _G. C. Greubel_, Oct 29 2018 *)
%o (PARI) m=80; x='x+O('x^m); Vec(1/(prod(k=1,2*m, prod(j=1,2*m, 1+x^(j*k) )))) \\ _G. C. Greubel_, Oct 29 2018
%o (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[(&*[1 + x^(j*k): j in [1..2*m]]): k in [1..2*m]]))); // _G. C. Greubel_, Oct 29 2018
%Y Cf. A107742, A109386.
%Y Product_{k>=1} 1/(1 + x^k)^sigma_m(k): this sequence (m=0), A288421 (m=1), A288422 (m=2), A288423 (m=3).
%K sign
%O 0,8
%A _Seiichi Manyama_, Jun 04 2017
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