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A267314 Expansion of 2*x*(1 + 2*x - x^2)/((1 - x)*(1 + x^2)^2).

%I A267314 #19 Sep 08 2022 08:46:15
%S A267314 0,2,6,0,-8,2,14,0,-16,2,22,0,-24,2,30,0,-32,2,38,0,-40,2,46,0,-48,2,
%T A267314 54,0,-56,2,62,0,-64,2,70,0,-72,2,78,0,-80,2,86,0,-88,2,94,0,-96,2,
%U A267314 102,0,-104,2,110,0,-112,2,118,0,-120,2,126,0,-128,2,134,0,-136,2,142,0
%N A267314 Expansion of 2*x*(1 + 2*x - x^2)/((1 - x)*(1 + x^2)^2).
%H A267314 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2,2,-1,1).
%F A267314 G.f.: 2*x*(1 + 2*x - x^2)/((1 - x)*(1 + x^2)^2).
%F A267314 a(n) = Sum_{k = 0..n} (2k)*(-1)^((-sin((Pi*k)/2) + cos((Pi*k)/2) + 1)/2).
%F A267314 a(n) = Sum_{k = 0..n} A005843(k)*(-1)^A133872(k + 1).
%F A267314 a(n) = 1 - (-1)^(n*(n+1)/2) - (1+(-1)^n)*(-1)^(n/2)*n. [_Bruno Berselli_, Jan 19 2016]
%e A267314 a(0) = 0;
%e A267314 a(1) = 0 + 2 = 2;
%e A267314 a(2) = 0 + 2 + 4 = 6;
%e A267314 a(3) = 0 + 2 + 4 - 6 = 0;
%e A267314 a(4) = 0 + 2 + 4 - 6 - 8 = -8;
%e A267314 a(5) = 0 + 2 + 4 - 6 - 8 + 10 = 2;
%e A267314 a(6) = 0 + 2 + 4 - 6 - 8 + 10 + 12 = 14;
%e A267314 a(7) = 0 + 2 + 4 - 6 - 8 + 10 + 12 - 14 = 0;
%e A267314 a(8) = 0 + 2 + 4 - 6 - 8 + 10 + 12 - 14 - 16 = -16;
%e A267314 a(9) = 0 + 2 + 4 - 6 - 8 + 10 + 12 - 14 - 16 + 18 = 2, etc.
%t A267314 Table[Sum[(2k)*(-1)^((-sin((Pi k)/2)+cos((Pi k)/2)+1)/2), {k, 0, n}], {n, 0, 80}]
%t A267314 CoefficientList[Series[2 x (x^2 - 2 x - 1) / ((x - 1) (x^2 + 1)^2), {x, 0, 100}], x] (* _Vincenzo Librandi_, Jan 13 2016 *)
%t A267314 Table[1 - (-1)^(n (n + 1)/2) - (1 + (-1)^n) (-1)^(n/2) n, {n, 0, 80}] (* _Bruno Berselli_, Jan 19 2016 *)
%o A267314 (PARI) concat(0, Vec(2*x*(1+2*x-x^2)/((1-x)*(1+x^2)^2) + O(x^100))) \\ _Michel Marcus_, Jan 13 2016
%o A267314 (Magma) &cat [[-8*n,2,8*n+6,0]: n in [0..20]]; // _Bruno Berselli_, Jan 19 2016
%Y A267314 Cf. A005843, A077140, A137501.
%K A267314 sign,easy
%O A267314 0,2
%A A267314 _Ilya Gutkovskiy_, Jan 13 2016