A307060 - OEIS

%I #12 Jan 24 2024 10:09:40

%S 1,-1,1,-2,4,-7,12,-21,38,-68,120,-212,377,-670,1188,-2107,3740,-6638,

%T 11778,-20898,37084,-65808,116775,-207212,367696,-652478,1157815,

%U -2054524,3645730,-6469316,11479734,-20370656,36147506,-64143372,113821732,-201975429,358403220,-635982680,1128544452,-2002589998

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

%C Invert transform of A081362.

%H G. C. Greubel, <a href="/A307060/b307060.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(2 - Product_{k>=1} (1 - x^(2*k-1))).

%F a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).

%F From _G. C. Greubel_, Jan 24 2024: (Start)

%F G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)}.

%F G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)

%t nmax = 39; CoefficientList[Series[1/(2 - Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( 1/(2 - (&*[1-x^(2*j-1): j in [1..m+2]])) )); // _G. C. Greubel_, Jan 24 2024

%o (SageMath)

%o m=80;

%o def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )

%o def A307060_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( f(x) ).list()

%o A307060_list(m) # _G. C. Greubel_, Jan 24 2024

%Y Cf. A081362, A299208, A304969, A307058.

%Y Cf. A307057, A307058, A307059, A307062, A307063.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Mar 21 2019