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Expansion of (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).
8

%I #23 Jun 17 2023 22:56:34

%S 5,347,33865,3318287,325158125,31862177827,3122168268785,

%T 305940628162967,29979059391701845,2937641879758617707,

%U 287858925156952833305,28207237023501619046047,2764021369378001713679165,270845886962020666321511987,26540132900908647297794495425,2600662178402085414517539039527

%N Expansion of (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

%C Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence s_k.

%H J. Mc Laughlin, <a href="http://www.fq.math.ca/Papers1/48-1/McLaughlin.pdf">An identity motivated by an amazing identity of Ramanujan</a>, Fib. Q., 48 (No. 1, 2010), 34-38.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

%F G.f.: (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

%F a(n) = 17/12 + (-(17*sqrt(6) - 43)/(2*sqrt(6) + 5)^(2*n) + (17*sqrt(6) + 43)*(2 sqrt(6) + 5)^(2*n))/24. - _Bruno Berselli_, Mar 02 2016

%t CoefficientList[Series[(-7 x^2 + 148 x - 5)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[Simplify[17/12 + (-(17 Sqrt[6] - 43)/(2 Sqrt[6] + 5)^(2 n) + (17 Sqrt[6] + 43) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* _Bruno Berselli_, Mar 02 2016 *)

%o (PARI) Vec((-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

%o (Sage)

%o gf = (-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1)

%o print(taylor(gf, x, 0, 20).list()) # _Bruno Berselli_, Mar 02 2016

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1))); // _Bruno Berselli_, Mar 02 2016

%Y Cf. A261004, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269555.

%K nonn,easy

%O 0,1

%A _Michel Marcus_, Feb 29 2016