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%I #10 Mar 30 2021 01:52:40
%S 0,-1,1,40,-81,-1559,4880,59039,-259119,-2161480,12785359,75835321,
%T -600035040,-2509213121,27110649761,75767088200,-1187303728401,
%U -1919146887799,50598599752240,28086422647519,-2102629012489359,951085683941080,85256703828122639
%N Expansion of x/(1 + x + 41*x^2).
%H G. C. Greubel, <a href="/A141528/b141528.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-41).
%F a(n) = (-1)^(n-1)*(p^n - q^n)/(p-q), where p = (1 + sqrt(163)*i)/2, q = (1 - sqrt(163)*i)/2.
%F G.f.: x/(1 + x + 41*x^2). - _Roger L. Bagula_, Apr 18 2010
%F a(n) = -a(n-1) -41*a(n-2), with a(0) = 0, a(1) = -1. - _G. C. Greubel_, Mar 29 2021
%t p:= (1 +Sqrt[163]*I)/2; q:= (1 -Sqrt[163]*I)/2; f[n_]:= (-1)^(n-1)*(p^n -q^n)/(p-q); Table[Simplify[f[n]], {n, 0, 30}] (* modified by _G. C. Greubel_, Mar 29 2021 *)
%t CoefficientList[Series[x/(1+x+41*x^2), {x,0,30}], x] (* _Roger L. Bagula_, Apr 18 2010; modified by _G. C. Greubel_, Mar 29 2021 *)
%t LinearRecurrence[{-1,-41}, {0,-1}, 30] (* _G. C. Greubel_, Mar 29 2021 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 30);
%o Coefficients(R!( x/(1+x+41*x^2) )); // _G. C. Greubel_, Mar 29 2021
%o (Sage)
%o def A141528_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( x/(1+x+41*x^2) ).list()
%o a=A141528_list(31); a[1:] # _G. C. Greubel_, Mar 29 2021
%Y Cf. A005846, A141527.
%K sign
%O 1,4
%A _Roger L. Bagula_, Aug 11 2008
%E Edited by _G. C. Greubel_, Mar 29 2021
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