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%I #25 Sep 04 2022 21:43:56
%S 1,3,2,-6,-20,-28,-8,56,144,176,32,-352,-832,-960,-128,1920,4352,4864,
%T 512,-9728,-21504,-23552,-2048,47104,102400,110592,8192,-221184,
%U -475136,-507904,-32768,1015808,2162688,2293760,131072,-4587520,-9699328
%N Binomial transform of [1, 2, -3, -4, 5, 6, -7, -8, 9, 10, ...].
%H Vincenzo Librandi, <a href="/A140230/b140230.txt">Table of n, a(n) for n = 1..206</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-8,8,-4).
%F A007318 * [1, 2, -3, -4, 5, 6, -7, -8, 9, 10, ...]; i.e., (+) signs when n == 1 or 2 (mod 4); (-) otherwise.
%F a(1+4*n) + a(2+4*n) + a(3+4*n) + a(4+4*n) = 0. - _Paul Curtz_, Apr 22 2011
%F a(n) = 4*a(n-1) - 8*a(n-2) + 8*a(n-3) - 4*a(n-4). - _Joerg Arndt_, Apr 25 2011
%F G.f.: x*(x-1)*(2*x^2-1) / (1-2*x+2*x^2)^2. - _R. J. Mathar_, Jun 02 2011
%F a(n) = Sum_{k=0..n-1} binomial(n-1,k)*(n-k)*(-1)^floor((n-k-1)/2). - _Wesley Ivan Hurt_, Sep 04 2022
%e a(4) = -6 = (1, 3, 3, 1) dot (1, 2, -3, -4) = (1 + 6 - 9 - 4).
%o (PARI) Vec((2*x^3-2*x^2-x+1)/(4*x^4-8*x^3+8*x^2-4*x+1)+O(x^66)) /* Joerg Arndt, Apr 25 2011 */
%K sign,easy
%O 1,2
%A _Gary W. Adamson_, May 13 2008
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