A147988 - OEIS

%I #13 Jul 01 2015 02:08:58

%S 1,1,0,1,0,1,0,1,0,4,0,4,0,1,0,1,0,11,0,45,0,88,0,88,0,45,0,11,0,1,0,

%T 1,0,26,0,293,0,1896,0,7866,0,22122,0,43488,0,60753,0,60753,0,43488,0,

%U 22122,0,7866,0,1896,0,293,0,26,0,1,0,1,0,57,0,1512,0,24858,0,284578,0

%N Coefficients of denominator polynomials Q(n,x) associated with reciprocation.

%C 1. Q(n,1)=A073834(n) for n>=1.

%C 2. For n>=3, Q(n)=Q(n,x)=i*T(n,i*x), where T(n) is the polynomial at A147986.

%C Thus all the zeros of Q(n,x), for n>=2, are nonreal.

%H 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 The basic idea is to iterate the reciprocation-sum mapping x/y -> x/y+y/x.

%F Let x be an indeterminate, P(1)=x, Q(1)=1 and for n>1, define P(n)=P(n-1)^2+Q(n-1)^2 and Q(n)=P(n-1)*Q(n-1), so that P(n)/Q(n)=P(n-1)/Q(n-1)-Q(n-1)/P(n-1).

%e Q(1) = 1

%e Q(2) = x

%e Q(3) = x^3+x

%e Q(4) = x^7+4*x^5+4*x^3+1

%e so that, as an array, the sequence begins with:

%e 1

%e 1 0

%e 1 0 1 0

%e 1 0 4 0 4 0 1

%Y Cf. A147985, A147986, A147987, A147989, A147990, A147991, A147992, A147993.

%K nonn,tabf

%O 1,10

%A _Clark Kimberling_, Nov 24 2008