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A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

%I A307059 #20 Jan 24 2024 10:09:01
%S A307059 1,-1,0,1,-1,1,-1,1,0,-2,4,-4,1,3,-5,4,-3,3,-1,-6,13,-12,2,9,-13,10,
%T A307059 -6,6,-4,-9,28,-30,5,25,-28,5,9,7,-27,11,32,-47,2,51,-27,-74,128,-34,
%U A307059 -131,183,-78,-15,-37,97,89,-480,649,-242,-498,904,-663,223,-140,169,488,-1818
%N A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).
%C A307059 Invert transform of A010815.
%C A307059 Alternating row sums of Riordan triangle (1, 1 - Product_{j>=1} (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - _Wolfdieter Lang_, Feb 17 2021
%H A307059 G. C. Greubel, <a href="/A307059/b307059.txt">Table of n, a(n) for n = 0..2500</a>
%F A307059 a(0) = 1; a(n) = Sum_{k=1..n} A010815(k)*a(n-k).
%F A307059 G.f.: 1/(2 - QPochhammer(x)). - _G. C. Greubel_, Sep 08 2023
%t A307059 nmax=65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, nmax}]), {x, 0, nmax}], x]
%o A307059 (Magma)
%o A307059 m:=80;
%o A307059 R<x>:=PowerSeriesRing(Integers(), m);
%o A307059 Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // _G. C. Greubel_, Sep 08 2023
%o A307059 (SageMath)
%o A307059 from sage.modular.etaproducts import qexp_eta
%o A307059 m=80;
%o A307059 def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )
%o A307059 def A307059_list(prec):
%o A307059     P.<x> = PowerSeriesRing(QQ, prec)
%o A307059     return P( f(x) ).list()
%o A307059 A307059_list(m) # _G. C. Greubel_, Sep 08 2023
%Y A307059 Cf. A010815, A055887, A299105, A341418.
%Y A307059 Cf. A307057, A307058, A307060, A307062, A307063.
%K A307059 sign
%O A307059 0,10
%A A307059 _Ilya Gutkovskiy_, Mar 21 2019