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%I #28 Sep 08 2022 08:46:00
%S 1,-1,3,3,-5,35,-21,51,187,-253,1035,-45,91,8099,-8277,25203,23035,
%T -38845,286539,-179949,442267,1490147,-2045205,8563635,-732869,
%U 1498499,65544843,-68410797,211488475,176048675,-300358101,2344363251,-1536690053,3822551747,11858974155
%N Expansion of 1 / ((1 - 2*x) * (1 + 3*x + 4*x^2)) in powers of x.
%H G. C. Greubel, <a href="/A200562/b200562.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Rec"order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,8).
%F a(n) = -a(n-1) + 2*a(n-2) + 8*a(n-3) for all n in Z.
%F 7*a(n) = 2^(n+1) +(-1)^n*( 5*A049072(n) -4*A049072(n-1) ). - _R. J. Mathar_, Nov 19 2011
%F a() = a(-n-3) * 2^(2*n+3) for all n in Z. - _Michael Somos_, Sep 17 2014
%F 0 = a(n)*(+4*a(n+1) + 2*a(n+2)) + a(n+1)*(+a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Sep 17 2014
%e G.f. = 1 - x + 3*x^2 + 3*x^3 - 5*x^4 + 35*x^5 - 21*x^6 + 51*x^7 + 187*x^8 + ...
%t LinearRecurrence[{-1,2,8},{-1,3,3},40] (* _Harvey P. Dale_, Aug 03 2012 *)
%t CoefficientList[Series[1/((1-2*x)*(1+3*x+4*x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 13 2018 *)
%o (PARI) {a(n) = if( n<0, polcoeff( x^3 / ((2 - x) * (4 + 3*x + x^2)) + x * O(x^-n), -n), polcoeff( 1 / ((1 - 2*x) * (1 + 3*x + 4*x^2)) + x * O(x^n), n))}; /* _Michael Somos_, Sep 17 2014 */
%o (PARI) x='x+O('x^50); Vec(1/((1-2*x)*(1+3*x+4*x^2))) \\ _G. C. Greubel_, Aug 13 2018
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1+3*x+4*x^2)))); // _G. C. Greubel_, Aug 13 2018
%K sign,easy
%O 0,3
%A _Krishnamurthy Balasubraniam_, Nov 19 2011
%E Definition from _R. J. Mathar_, Nov 19 2011
%E Added a(0) = 1. - _Michael Somos_, Sep 17 2014
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