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A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

%I A307060 #15 Mar 17 2025 02:39:45
%S A307060 1,-1,1,-2,4,-7,12,-21,38,-68,120,-212,377,-670,1188,-2107,3740,-6638,
%T A307060 11778,-20898,37084,-65808,116775,-207212,367696,-652478,1157815,
%U A307060 -2054524,3645730,-6469316,11479734,-20370656,36147506,-64143372,113821732,-201975429,358403220,-635982680,1128544452,-2002589998
%N A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).
%C A307060 Invert transform of A081362.
%H A307060 G. C. Greubel, <a href="/A307060/b307060.txt">Table of n, a(n) for n = 0..1000</a>
%F A307060 G.f.: 1/(2 - Product_{k>=1} (1 - x^(2*k-1))).
%F A307060 a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).
%F A307060 From _G. C. Greubel_, Jan 24 2024: (Start)
%F A307060 G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)).
%F A307060 G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)
%t A307060 nmax = 39; CoefficientList[Series[1/(2 - Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%o A307060 (Magma)
%o A307060 m:=80;
%o A307060 R<x>:=PowerSeriesRing(Integers(), m);
%o A307060 Coefficients(R!( 1/(2 - (&*[1-x^(2*j-1): j in [1..m+2]])) )); // _G. C. Greubel_, Jan 24 2024
%o A307060 (SageMath)
%o A307060 m=80;
%o A307060 def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )
%o A307060 def A307060_list(prec):
%o A307060     P.<x> = PowerSeriesRing(QQ, prec)
%o A307060     return P( f(x) ).list()
%o A307060 A307060_list(m) # _G. C. Greubel_, Jan 24 2024
%Y A307060 Cf. A081362, A299208, A304969, A307058.
%Y A307060 Cf. A307057, A307058, A307059, A307062, A307063.
%K A307060 sign
%O A307060 0,4
%A A307060 _Ilya Gutkovskiy_, Mar 21 2019