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%I #34 Sep 08 2022 08:45:48
%S 1,1,0,-1,-1,-2,-2,-3,-3,-4,-4,-5,-5,-6,-6,-7,-7,-8,-8,-9,-9,-10,-10,
%T -11,-11,-12,-12,-13,-13,-14,-14,-15,-15,-16,-16,-17,-17,-18,-18,-19,
%U -19,-20,-20,-21,-21,-22,-22,-23,-23,-24,-24,-25,-25,-26,-26,-27,-27
%N Hankel transform of A168049.
%H G. C. Greubel, <a href="/A168050/b168050.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1)
%F G.f.: (1 - 2x^2 - x^3 + x^4)/((1+x)(1-x)^2).
%F a(n) = + 1*a(n-1) + 1*a(n-2) - 1*a(n-3). - _Joerg Arndt_, Apr 02 2011
%F a(n) = (-1)^n/4 -(2n-3)/4 + C(1,n) - C(0,n).
%F E.g.f.: (4*x + exp(-x) - (2*x - 3)*exp(x))/4. - _Ilya Gutkovskiy_, Jul 08 2016
%t Join[{1,1,b=0},a=0;Table[c=b+2*a+n;a=b;b=c,{n,-1,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2011 *)
%t CoefficientList[Series[(1 - 2 x^2 - x^3 + x^4)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* _G. C. Greubel_, Jul 07 2016 *)
%t Table[(-1)^n/4 - (2 n - 3)/4 + Binomial[1, n] - Binomial[0, n], {n, 0, 80}] (* _Vincenzo Librandi_, Jul 08 2016 *)
%t LinearRecurrence[{1,1,-1},{1,1,0,-1,-1},60] (* _Harvey P. Dale_, Dec 05 2018 *)
%o (Magma) [(-1)^n/4-(2*n-3)/4+Binomial(1,n)-Binomial(0,n): n in [0..80]]; // _Vincenzo Librandi_, Jul 08 2016
%o (PARI) Vec((1-2*x^2-x^3+x^4)/((1+x)*(1-x)^2) + O(x^99)) \\ _Altug Alkan_, Jul 08 2016
%Y Cf. A168049.
%K easy,sign
%O 0,6
%A _Paul Barry_, Nov 17 2009
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