%I #33 Sep 08 2022 08:46:19

%S 1,-1,-1,-1,-2,-4,-9,-21,-51,-127,-323,-835,-2188,-5798,-15511,-41835,

%T -113634,-310572,-853467,-2356779,-6536382,-18199284,-50852019,

%U -142547559,-400763223,-1129760415,-3192727797,-9043402501,-25669818476,-73007772802

%N Expansion of (1 - x + sqrt(1 - 2*x - 3*x^2))/2 in powers of x.

%C Apart from a(1) the same as A168051. - _R. J. Mathar_, Sep 18 2017

%H G. C. Greubel, <a href="/A292440/b292440.txt">Table of n, a(n) for n = 0..1000</a>

%F Convolution inverse of A001006.

%F Let f(x) = (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2).

%F G.f.: 1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(... (continued fraction).

%F G.f.: 1/f(x) = 1 - x - x^2*f(x).

%F a(n) = -A001006(n-2) for n > 1.

%F a(n) ~ -3^(n - 1/2) / (2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 14 2018

%F D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - _R. J. Mathar_, Jan 23 2020

%t CoefficientList[Series[(1-x +Sqrt[1-2*x-3*x^2])/2, {x, 0, 50}], x] (* _G. C. Greubel_, Aug 13 2018 *)

%o (PARI) x='x+O('x^50); Vec((1 - x + sqrt(1 - 2*x - 3*x^2))/2) \\ _G. C. Greubel_, Aug 13 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x +Sqrt(1-2*x-3*x^2))/2)); // _G. C. Greubel_, Aug 13 2018

%Y Cf. A001006, A086246.

%K sign

%O 0,5

%A _Seiichi Manyama_, Sep 16 2017