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%I #13 Apr 02 2019 05:53:55
%S 1,-1,-2,0,0,3,1,3,1,-2,0,-3,-6,-4,1,-8,1,2,5,5,4,9,13,7,3,1,3,7,-16,
%T -9,-17,-13,-21,-5,-25,-33,-3,-3,-9,22,-6,11,29,29,57,37,40,31,58,18,
%U 35,40,37,-24,-36,-34,-29,-60,-54,-98,-74,-124,-113,-156,-71,-35,-140,-46,-16,-61,-25
%N Expansion of Product_{k>=1} (1 - x^k - x^(2*k)).
%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k)))/(j*k)), where phi = Euler totient function (A000010).
%p a:=series(mul((1-x^k-x^(2*k)),k=1..100),x=0,71): seq(coeff(a,x,n),n=0..70); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 70; CoefficientList[Series[Product[(1 - x^k - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 70; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Sum[EulerPhi[d/j] (Fibonacci[j - 1] + Fibonacci[j + 1]), {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]
%Y Cf. A000010, A000358, A000726, A109389, A137569, A162891, A263401.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Sep 25 2018
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