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A147986 Coefficients of denominator polynomials T(n,x) associated with reciprocation.

%I A147986 #17 Jul 01 2015 02:08:33
%S A147986 1,1,0,1,0,-1,0,1,0,-4,0,4,0,-1,0,1,0,-11,0,45,0,-88,0,88,0,-45,0,11,
%T A147986 0,-1,0,1,0,-26,0,293,0,-1896,0,7866,0,-22122,0,43488,0,-60753,0,
%U A147986 60753,0,-43488,0,22122,0,-7866,0,1896,0,-293,0,26,0,-1,0,1,0,-57,0,1512,0
%N A147986 Coefficients of denominator polynomials T(n,x) associated with reciprocation.
%C A147986 T(n)=S(1)*S(2)*...*S(n-1). The degree of S(n) in x is m=2^(n-1), so that the degree of T(n) is m-1. Write the zeros of T(n) as r(1)<r(2)<...<r(m-1) and the zeros of S(n) as z(1)<z(2)<...<z(m). Then z(1)<r(1)<z(2)<r(2)<...<r(m-1)<z(m); i.e., the zeros of T(n) intersperse the zeros of S(n).
%H A147986 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.
%F A147986 The basic idea is to iterate the reciprocation-difference mapping x/y -> x/y-y/x.
%F A147986 Let x be an indeterminate, S(1)=x, T(1)=1 and for n>1, define S(n)=S(n-1)^2-T(n-1)^2 and T(n)=S(n-1)*T(n-1), so that S(n)/T(n)=S(n-1)/T(n-1)-T(n-1)/S(n-1).
%e A147986 T(1) = 1
%e A147986 T(2) = x
%e A147986 T(3) = x^3-x
%e A147986 T(4) = x^7-4*x^5+4*x^3-x
%e A147986 so that, as an array, the sequence begins with:
%e A147986 1
%e A147986 1 0
%e A147986 1 0 -1 0
%e A147986 1 0 -4 0 4 0 -1 0
%t A147986 s[1] = x; t[1] = 1; s[n_] := s[n] = s[n-1]^2 - t[n-1]^2; t[n_] := t[n] = s[n-1]*t[n-1]; row[n_] := CoefficientList[t[n], x] // Reverse; Table[row[n], {n, 7}] // Flatten (* _Jean-François Alcover_, Apr 22 2013 *)
%Y A147986 Cf. A147985, A147987, A147988, A147989, A147990, A147991, A147992, A147993.
%K A147986 sign,tabf
%O A147986 1,10
%A A147986 _Clark Kimberling_, Nov 24 2008