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Expansion of (1 - x^2) / (1 + x^2) in powers of x.
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%I #50 Oct 13 2024 17:41:45

%S 1,0,-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 -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 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,-2,0,2

%N Expansion of (1 - x^2) / (1 + x^2) in powers of x.

%C Difference sequence of A057077.

%C Sequence of determinants of matrices for some bipartite graphs, called Tz(n). The graph Tz(4) appears in the logo for the beer called Tannenzäpfle (small fir cone), brewed by Badische Staatsbrauerei Rothaus, Germany, hence the name Tz. See the link for this logo with Tz(4).

%C The vertex-vertex matrix for these bipartite graphs will also be called Tz(n) (without leading to confusion).

%C General proof by expanding the determinant a(n) = determinant(Tz(n)) along the first column yielding b(n-1)-b(n-2), with b(n-1) the A_{1,1} minor of the matrix Tz(n), and deriving a recurrence for the b(n), namely b(n) = -b(n-2) with inputs b(0) = 1 = b(1). This gives b(n) = A057077(n), n>=0.

%H Wolfdieter Lang, <a href="/A176742/a176742.pdf">Some Tz(n) graphs and matrices.</a>

%H Rothaus Tannenzäpfle, <a href="https://www.rothaus.de/biere/tannenzaepfle">Tz(4) graph on the Tannenzäpfle logo</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1).

%F Euler transform of length 4 sequence [0, -2, 0, 1]. - _Michael Somos_, Mar 21 2011

%F Moebius transform is length 4 sequence [0, -2, 0, 4]. - _Michael Somos_, Mar 22 2011

%F a(n) = a(-n) for all n in Z. a(n) = c_4(n) if n>1, where c_k(n) is Ramanujan's sum. - _Michael Somos_, Mar 21 2011

%F a(n-1) := determinant(Tz(n)), n>=1. The rows of the matrix Tz(4) are [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]]. Tz(1)=(1), and Tz(2) has rows [[1, 1], [1, 1]]. The matrix for the generalization Tz(n) has rows [[1,1,0,...,0], [1,0,1,0,...,0], [0,1,0,1,0,...,0], ..., [0,...,0,1,0,1], [0,...,0,1,1].

%F a(0)=1, a(2*k-1)= 0, a(4*k) = +2, a(4*k-2) = -2, k>=1.

%F O.g.f.: (1-x^2)/(1+x^2).

%F a(n) = A057077(n) - A057077(n-1), n>=1. a(0)=1.

%F Dirichlet g.f. sum_{n>=1} a(n)/n^s = zeta(s)*(4^(1-s)-2^(1-s)). - _R. J. Mathar_, Apr 11 2011

%F a(n) = (((n+1) mod 2)+((n+2+sign(n)) mod 2))*(-1)^ceiling(n/2). - _Wesley Ivan Hurt_, Jun 20 2014

%e G.f. = 1 - 2*x^2 + 2*x^4 - 2*x^6 + 2*x^8 - 2*x^10 + 2*x^12 - 2*x^14 + 2*x^16 + ...

%e The bipartite graphs Tz(n) (n>=1) look like |, |X|, |XX|, |XXX|, ... For n>=2 the lines have to be connected to give the 2*n nodes and 2*n edges. The n=1 graph Tz(1) has 2*1=2 nodes and only one edge.

%e n=1: determinant((1))=1; n=2: determinant(Matrix([[1,1],[1,1]]))=0; n=3: determinant(Matrix([[1,1,0],[1,0,1],[0,1,1]]))=-2; n=4: determinant(Tz(4))=0; etc.

%p a := n -> 2^signum(n)*(-1)^iquo(n+1,2)*modp(n+1,2);

%p seq(a(n), n=0..100); # _Peter Luschny_, Jun 22 2014

%t Join[{1}, Table[{0, -2, 0, 2}, {26}]] // Flatten (* _Jean-François Alcover_, Jun 21 2013 *)

%t a[ n_] := - Boole[n == 0] + {0, -2, 0, 2}[[Mod[ n, 4, 1]]]; (* _Michael Somos_, May 05 2015 *)

%t PadRight[{1},120,{2,0,-2,0}] (* _Harvey P. Dale_, Apr 13 2019 *)

%o (PARI) {a(n) = - (n == 0) + [2, 0, -2, 0][n%4 + 1]}; /* _Michael Somos_, Mar 21 2011 */

%Y Cf. A084099, A057077.

%K sign,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 15 2010