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A transform of the Fibonacci numbers.
1

%I #18 Mar 22 2017 03:37:21

%S 1,1,-1,3,-13,65,-353,2025,-12077,74143,-465481,2974863,-19289821,

%T 126594191,-839273105,5612483619,-37814455781,256447068841,

%U -1749182184793,11991887667273,-82588248514885,571118483653841

%N A transform of the Fibonacci numbers.

%C A transform of F(n+1) by the inverse of the Riordan array (1, x*(1+x)/(1-2*x)).

%C Equivalently, row sums of the inverse of the Riordan array (1, x/(2-sqrt(1+4*x)).

%C Hankel transform is alternating sign version of A083667.

%H Vincenzo Librandi, <a href="/A141342/b141342.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: 1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2)).

%F Conjecture: (n-1)*a(n) +4*(n-4)*a(n-1) + (65-29*n)*a(n-2) +12*(7-2*n)*a(n-3)+ 4*(4-n)*a(n-4) =0. - _R. J. Mathar_, Nov 14 2011

%F a(n) ~ (-1)^n * (5*sqrt(3)-14) * sqrt(2*sqrt(3)-3) * 2^(n+1/2) * (2+sqrt(3))^n / (121 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2014

%t CoefficientList[Series[1/(1-2*x-2*x^2+x*Sqrt[1+8*x+4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%o (PARI) x='x+O('x^50); Vec(1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2))) \\ _G. C. Greubel_, Mar 21 2017

%Y Cf. A141343.

%K easy,sign

%O 0,4

%A _Paul Barry_, Jun 26 2008