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A243816 Expansion of (x*sqrt(4*x^2+1)-x)/(x*sqrt(-(2*sqrt(4*x^2+1)-x-2)/x) + sqrt(4*x^2+1)-x-1). 1

%I #21 Sep 08 2022 08:46:08

%S -1,2,0,2,5,10,27,86,264,806,2559,8332,27343,90498,302801,1022074,

%T 3472577,11868242,40786623,140851104,488490057,1700694884,5941890068,

%U 20826229564,73208513161,258031793698,911704655945

%N Expansion of (x*sqrt(4*x^2+1)-x)/(x*sqrt(-(2*sqrt(4*x^2+1)-x-2)/x) + sqrt(4*x^2+1)-x-1).

%H G. C. Greubel, <a href="/A243816/b243816.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (Sum_{i=0..n/2} (-1)^i*binomial(2*(n-1)-4*i, n-2*i)*binomial(n-1, i))/(n-1), n > 1, a(0)=-1, a(1)=2.

%F a(n) ~ 4*(15/4)^n / (sqrt(255*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 15 2014

%F Conjecture D-finite with recurrence: 2*n*(n-1)*(2*n-3)*a(n) -(n-1)*(47*n^2-221*n+260)*a(n-1) +4*(38*n^3-300*n^2+777*n-660)*a(n-2) +4*(-94*n^3+1006*n^2-3617*n+4355)*a(n-3) +16*(n-5)*(64*n^2-510*n+1041)*a(n-4) -16*(n-5)*(n-6)*(47*n-221)*a(n-5) +1920*(n-5)*(n-6)*(n-7)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[x*(-1 + Sqrt[1 + 4*x^2])/ (-1 + Sqrt[1 + 4*x^2] + x*(-1 + Sqrt[(2 + x - 2*Sqrt[1 + 4*x^2])/x])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)

%o (Maxima)

%o a(n):=if n=0 then -1 else if n=1 then 2 else sum((-1)^i*binomial(2*(n-1)-4*i,n-2*i)*binomial(n-1,i),i,0,n/2)/(n-1);

%o (PARI) x='x+O('x^30); Vec((x*sqrt(4*x^2+1)-x)/(x*sqrt(-(2*sqrt(4*x^2+1)-x-2)/x)+sqrt(4*x^2+1)-x-1)) \\ _G. C. Greubel_, Oct 06 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((x*Sqrt(4*x^2+1)-x)/(x*Sqrt(-(2*Sqrt(4*x^2+1)-x-2)/x)+Sqrt(4*x^2+1)-x-1))); // _G. C. Greubel_, Oct 06 2018

%K sign

%O 0,2

%A _Vladimir Kruchinin_, Jun 11 2014

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