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%I #12 Jan 25 2024 04:56:23
%S 1,0,1,1,2,4,6,12,19,36,60,108,187,328,576,1005,1765,3084,5408,9461,
%T 16575,29017,50812,88977,155792,272813,477684,836466,1464654,2564685,
%U 4490833,7863610,13769463,24110774,42218847,73926591,129448088,226667986,396903536,694991728
%N Expansion of 1/(2 - Product_{k>=2} (1 + x^k)).
%C Invert transform of A025147.
%H G. C. Greubel, <a href="/A307067/b307067.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0) = 1; a(n) = Sum_{k=1..n} A025147(k)*a(n-k).
%F From _G. C. Greubel_, Jan 24 2024: (Start)
%F G.f.: (1+x)/(2*(1+x) - QP(x^2)/QP(x)), where QP(x) = QPochhammer(x).
%F G.f.: (1+x)/(2*(1+x) - x^(1/24)*eta(x^2)/eta(x)), where eta(x) is the Dedekind eta function. (End)
%p a:=series(1/(2-mul((1+x^k),k=2..100)),x=0,40): seq(coeff(a,x,n),n=0..39); # _Paolo P. Lava_, Apr 03 2019
%t nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
%o (Magma)
%o m:=80;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( 1/(2 - (&*[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 - product(1+x^j for j in range(2, m+3)) )
%o def A307067_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(x) ).list()
%o A307067_list(m) # _G. C. Greubel_, Jan 24 2024
%Y Cf. A025147, A055887, A299106, A304969, A307057, A307060, A317536.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Mar 22 2019
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