A143471 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.

1, 0, -10, 1, 55, -12, -219, 77, 701, -351, -1900, 1277, 4494, -3966, -9485, 11058, 18342, -29012, -34057, 75053, 65836, -198845, -144194, 547462, 359314, -1548522, -937883, 4396415, 2346732, -12282817, -5272447, 33415657, 9836296, -88340069, -11914604

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,-10,1,-45,8,-120,27,-210,48,-253,48,-210,27,-120,8,-45,1,-10,0,-1).

Formula

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).

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

Mathematica

f[x_] = x^10 - x^7 - x^5 - x^3 + 1;
CoefficientList[Series[1/(x^10*f[x + 1/x]), {x, 0, 40}], x]
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 *)

Prog

(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
(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

Crossrefs

Cf. A143471, A143478, A167603.

Sequence in context: A049326 A146537 A050304 * A286921 A009209 A009227

Adjacent sequences: A143468 A143469 A143470 * A143472 A143473 A143474

Keyword

easy,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 24 2008

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 30 2018

Status

approved