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Expansion of 1/(x^10*p(x + 1/x)), where p(x) = 1 - x^3 - x^5 - x^7 + x^10 is a Salem polynomial.
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%I #17 Dec 20 2023 15:49:06

%S 1,0,-10,1,55,-12,-219,77,701,-351,-1900,1277,4494,-3966,-9485,11058,

%T 18342,-29012,-34057,75053,65836,-198845,-144194,547462,359314,

%U -1548522,-937883,4396415,2346732,-12282817,-5272447,33415657,9836296,-88340069,-11914604

%N Expansion of 1/(x^10*p(x + 1/x)), where p(x) = 1 - x^3 - x^5 - x^7 + x^10 is a Salem polynomial.

%H G. C. Greubel, <a href="/A143471/b143471.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,-10,1,-45,8,-120,27,-210,48,-253,48,-210,27,-120,8,-45,1,-10,0,-1).

%F G.f.: 1/(1 + 10*x^2 - x^3 + 45*x^4 - 8*x^5 + 120*x^6 - 27*x^7 + 210*x^8 - 48*x^9 + 253*x^10 - 48*x^11 + 210*x^12 - 27*x^13 + 120*x^14 - 8*x^15 + 45*x^16 - x^17 + 10*x^18 + x^20).

%F a(n) = -10*a(n-2) + a(n-3) - 45*a(n-4) + 8*a(n-5) - 120*a(n-6) + 27*a(n-7) - 210*a(n-8) + 48*a(n-9) - 253*a(n-10) + 48*a(n-11) - 210*a(n-12) + 27*a(n-13) - 120*a(n-14) + 8*a(n-15) - 45*a(n-16) + a(n-17) - 10*a(n-18) - a(n-20). - _Franck Maminirina Ramaharo_, Oct 30 2018

%t f[x_] = x^10 - x^7 - x^5 - x^3 + 1;

%t CoefficientList[Series[1/(x^10*f[x + 1/x]), {x, 0, 40}], x]

%t LinearRecurrence[{0,-10,1,-45,8,-120,27,-210,48,-253,48,-210,27,-120,8,-45,1,-10,0,-1},{1,0,-10,1,55,-12,-219,77,701,-351,-1900,1277,4494,-3966,-9485,11058,18342,-29012,-34057,75053},40] (* _Harvey P. Dale_, Dec 20 2023 *)

%o (PARI) x='x+O('x^50); Vec(1/(1+10*x^2-x^3+45*x^4-8*x^5+120*x^6-27*x^7 + 210*x^8-48*x^9+253*x^10-48*x^11+210*x^12-27*x^13+120*x^14 - 8*x^15 + 45*x^16-x^17+10*x^18+x^20)) \\ _G. C. Greubel_, Nov 03 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + 10*x^2-x^3+45*x^4-8*x^5+120*x^6-27*x^7+210*x^8-48*x^9+253*x^10 - 48*x^11+210*x^12-27*x^13+120*x^14-8*x^15+45*x^16-x^17+10*x^18+x^20))); // _G. C. Greubel_, Nov 03 2018

%Y Cf. A143471, A143478, A167603.

%K easy,sign

%O 0,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 24 2008

%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 30 2018