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%I #61 Mar 07 2023 19:04:34
%S 0,1,0,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,
%T -21,22,-23,24,-25,26,-27,28,-29,30,-31,32,-33,34,-35,36,-37,38,-39,
%U 40,-41,42,-43,44,-45,46,-47,48,-49,50,-51,52,-53,54,-55,56,-57,58,-59,60
%N Expansion of x*(1+2*x)/(1+x)^2.
%C Partial sums of A097140.
%C Binomial transform is x(1+x)/(1-x), or {0,1,2,2,2,2,....}.
%C Second binomial transform is x/((1-x)^2(1 - 2x)), or Eulerian numbers A000295(n+1).
%H Vincenzo Librandi, <a href="/A097141/b097141.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1).
%F G.f.: x*(1+2*x)/(1+x)^2.
%F a(n) = (n-2)*(-1)^n + 2*0^n.
%F a(n) = -2*a(n-1) - a(n-2) for n > 2.
%F a(n) = A099570(n) for n > 1. - _R. J. Mathar_, Dec 15 2008
%F a(n) = (Sum_{k=1..n} k*(-1)^(n-k)*binomial(n-1,k-1)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=0. - _Vladimir Kruchinin_, Mar 09 2014
%F a(n) = A038608(n-2) for n > 2. - _Georg Fischer_, Oct 06 2018
%F E.g.f.: 2 - exp(-x)*(2 + x). - _Stefano Spezia_, Mar 07 2023
%p A097141:=n->(n-2)*(-1)^n: 0, seq(A097141(n), n=1..100); # _Wesley Ivan Hurt_, Dec 11 2016
%t CoefficientList[Series[x (1 + 2 x)/(1 + x)^2, {x, 0, 100}], x] (* _Vincenzo Librandi_, Mar 11 2014 *)
%o (Magma) [0] cat [(n-2)*(-1)^n : n in [1..100]]; // _Wesley Ivan Hurt_, Dec 11 2016
%o (PARI) a(n)=if(n, (n-2)*(-1)^n, 0) \\ _Charles R Greathouse IV_, Dec 13 2016
%Y Cf. A000295, A038608, A040000, A097140, A099570.
%K easy,sign
%O 0,5
%A _Paul Barry_, Jul 29 2004
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