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Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).
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%I #16 Sep 08 2022 08:46:23

%S 1,1,1,0,1,-2,5,-12,28,-63,137,-290,604,-1253,2617,-5537,11870,-25666,

%T 55617,-120103,257582,-548119,1158437,-2437114,5117165,-10748530,

%U 22621055,-47728657,100932549,-213750621,452855190,-958925784,2028187595,-4283531490,9033779224

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

%H G. C. Greubel, <a href="/A320590/b320590.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: exp(Sum_{k>=1} x^k/(k*((1 + x)^k - x^k))).

%F G.f.: exp(Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k)).

%p seq(coeff(series(mul(1/(1-x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Oct 16 2018

%t nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1 - x^k/(1 + x)^k))) \\ _G. C. Greubel_, Oct 29 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1 - x^k/(1 + x)^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Oct 29 2018

%Y Cf. A000203, A103446, A218482, A320568, A320591.

%K sign

%O 0,6

%A _Ilya Gutkovskiy_, Oct 16 2018