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A084099 Expansion of (1+x)^2/(1+x^2).

%I A084099 #45 May 16 2025 00:56:19
%S A084099 1,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,
%T A084099 -2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,
%U A084099 2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,0
%N A084099 Expansion of (1+x)^2/(1+x^2).
%C A084099 Inverse binomial transform of A077860. Partial sums of A084100.
%C A084099 Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - _Paul Barry_, Oct 12 2004
%C A084099 Euler transform of length 4 sequence [2, -3, 0, 1]. - _Michael Somos_, Aug 04 2009
%H A084099 Colin Barker, <a href="/A084099/b084099.txt">Table of n, a(n) for n = 0..1000</a>
%H A084099 Bérénice Delcroix-Oger and Clément Dupont, <a href="https://arxiv.org/abs/2505.06094">Lie-operads and operadic modules from poset cohomology</a>, arXiv:2505.06094 [math.CO], 2025. See p. 28.
%H A084099 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1).
%F A084099 G.f.: (1+x)^2/(1+x^2).
%F A084099 a(n) = 2 * A101455(n) for n>0. - _N. J. A. Sloane_, Jun 01 2010
%F A084099 a(n+2) = (-1)^A180969(1,n)*((-1)^n - 1). - _Adriano Caroli_, Nov 18 2010
%F A084099 G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 19 2013
%F A084099 From _Wesley Ivan Hurt_, Oct 27 2015: (Start)
%F A084099 a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
%F A084099 a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
%F A084099 a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
%F A084099 From _Colin Barker_, Oct 27 2015: (Start)
%F A084099 a(n) = -a(n-2).
%F A084099 a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
%F A084099 (End)
%e A084099 G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...
%p A084099 A084099:=n->(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4): 1,seq(A084099(n), n=1..100); # _Wesley Ivan Hurt_, Oct 27 2015
%t A084099 CoefficientList[Series[(1+x)^2/(1+x^2),{x,0,110}],x] (* or *) Join[ {1}, PadRight[{},120,{2,0,-2,0}]] (* _Harvey P. Dale_, Nov 23 2011 *)
%o A084099 (PARI) {a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* _Michael Somos_, Aug 04 2009 */
%o A084099 (Magma) [1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // _Wesley Ivan Hurt_, Oct 27 2015
%o A084099 (PARI) a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ _Colin Barker_, Oct 27 2015
%o A084099 (PARI) Vec((1+x)^2/(1+x^2) + O(x^100)) \\ _Colin Barker_, Oct 27 2015
%Y A084099 Cf. A063886, A077860, A084100, A101455, A180969.
%K A084099 sign,easy
%O A084099 0,2
%A A084099 _Paul Barry_, May 15 2003