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%I #16 Jan 02 2023 02:18:55
%S 0,1,1,-1,-1,5,5,-15,-15,55,55,-197,-197,727,727,-2705,-2705,10165,
%T 10165,-38455,-38455,146301,146301,-559131,-559131,2145025,2145025,
%U -8255575,-8255575,31861025,31861025,-123256495,-123256495,477823895,477823895,-1855782325,-1855782325,7219352975
%N Expansion of x/((1-x)*sqrt(1+4*x^2)).
%C A transformation of the Fibonacci numbers A000045 by the Riordan array (1/sqrt(1+4*x^2), (sqrt(1+4*x^2)-1)/(2*x)).
%H G. C. Greubel, <a href="/A104551/b104551.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: x/((1-x)*sqrt(1+4*x^2)).
%F a(n) = Sum_{k=0..n} (sin(Pi*k/2)+cos(Pi*k)/2+1/2)*C(k-1,(k-1)/2)*(1-(-1)^k)/2.
%F D-finite with recurrence: (n-1)*a(n) = (n-1)*a(n-1) - 4*(n-2)*a(n-2) + 4*(n-2)*a(n-3). - _R. J. Mathar_, Feb 20 2015
%t CoefficientList[Series[x/((1-x)*Sqrt[1+4*x^2]), {x,0,40}], x] (* _G. C. Greubel_, Jan 01 2023 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*Sqrt(1+4*x^2)) )); // _G. C. Greubel_, Jan 01 2023
%o (SageMath)
%o def A104551_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x/((1-x)*sqrt(1+4*x^2)) ).list()
%o A104551_list(40) # _G. C. Greubel_, Jan 01 2023
%Y Cf. A000045, A054108.
%K easy,sign
%O 0,6
%A _Paul Barry_, Mar 14 2005
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