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A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2*k) - x^(3*k)). 1

%I

%S 1,1,3,6,13,24,51,93,184,343,654,1211,2286,4217,7865,14521,26912,

%T 49600,91669,168800,311305,573058,1055576,1942437,3575840,6578762,

%U 12106121,22270404,40972700,75367724,138644224,255020102,469095029,862827347,1587061299,2919111935,5369224903

%N Expansion of Product_{k>=1} 1/(1 - x^k - x^(2*k) - x^(3*k)).

%H G. C. Greubel, <a href="/A320286/b320286.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: exp(-Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k) + x^(2*j*k)))/(j*k)), where phi = Euler totient function (A000010).

%F From _Vaclav Kotesovec_, Oct 09 2018: (Start)

%F a(n) ~ s*p / r^(n+1), where

%F r = A192918 = ((17 + 3*sqrt(33))^(1/3) - 2/(17 + 3*sqrt(33))^(1/3) - 1)/3 = 0.54368901269207636157085597180174798652520329765098393524... is the real root of the equation 1 - r - r^2 - r^3 = 0,

%F s = (51 + 9*sqrt(33))/(4*(17 + 3*sqrt(33))^(1/3) + (17 + 3*sqrt(33))^(5/3) - 34 - 6*sqrt(33)) = 0.3362281169949410942253629540143324151579260900204592... is the real root of the equation -1 - 2*s + 44*s^3 = 0,

%F p = Product_{k>=2} 1/(1 - r^k - r^(2*k) - r^(3*k)) = 2.577933056783997593784130068093034525002002622982961271582417329674...

%F (End)

%p seq(coeff(series(mul(((1-x^k-x^(2*k)-x^(3*k)))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 40); # _Muniru A Asiru_, Oct 25 2018

%t nmax = 36; CoefficientList[Series[Product[1/(1 - x^k - x^(2 k) - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 36; CoefficientList[Series[Exp[-Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k) + x^(2 j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

%o (PARI) m=40; x='x+O('x^m); Vec(1/prod(k=1, m+2, (1-x^k-x^(2*k)-x^(3*k)))) \\ _G. C. Greubel_, Oct 24 2018

%o (MAGMA) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/( &*[(1-x^k-x^(2*k)-x^(3*k)): k in [1..m+2]]))); // _G. C. Greubel_, Oct 24 2018

%Y Cf. A000010, A001935, A082303, A093305, A162891.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 09 2018

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Last modified April 13 03:43 EDT 2021. Contains 342934 sequences. (Running on oeis4.)