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%I #11 Jan 24 2024 10:08:41
%S 1,0,1,1,3,4,10,15,33,55,111,195,380,684,1306,2389,4507,8313,15591,
%T 28881,53991,100257,187086,347860,648512,1206656,2248399,4185087,
%U 7796011,14514195,27033073,50334299,93741325,174552379,325067573,605316388,1127249250,2099115548,3909023438,7279285948
%N Expansion of 1/(2 - Product_{k>=2} 1/(1 - x^k)).
%C Invert transform of A002865.
%H G. C. Greubel, <a href="/A307057/b307057.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0) = 1; a(n) = Sum_{k=1..n} A002865(k)*a(n-k).
%F a(n) ~ c / r^n, where r = 0.53700045638650021831634004949965496126950171484122... is the root of the equation 1 - r = 2*QPochhammer[r] and c = 0.2143395760756683581919851351414497181589685708674442097498294834747517926...
%F From _G. C. Greubel_, Jan 24 2024: (Start)
%F G.f.: 1/( 2 - (1-x)/QPochhammer(x) ).
%F G.f.: 1/( 2 - x^(1/24)*(1-x)/eta(x) ), where eta(x) is the Dedekind eta function. (End)
%t nmax = 50; CoefficientList[Series[1/(2 - Product[1/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
%t a[0]= 1; a[n_]:= a[n]= Sum[(PartitionsP[k] -PartitionsP[k-1]) a[n-k], {k,n}];
%t Table[a[n], {n,0,50}]
%t CoefficientList[Series[1/(2 -(1-x)/QPochhammer[x]), {x,0,80}], x] (* _G. C. Greubel_, Jan 24 2024 *)
%o (Magma)
%o m:=80;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( 1/(2 - 1/(&*[1 - x^j: j in [2..m+2]])) )); // _G. C. Greubel_, Jan 24 2024
%o (SageMath)
%o m=80;
%o def f(x): return 1/( 2 - (1-x)/product(1 - x^j for j in range(1,m+3)) )
%o def A307057_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(x) ).list()
%o A307057_list(m) # _G. C. Greubel_, Jan 24 2024
%Y Cf. A002865, A055887, A304969, A307058, A307059, A307060, A307062, A307063.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Mar 21 2019
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