A147987 - OEIS

%I #16 Jul 01 2015 02:08:47

%S 1,0,1,0,1,1,0,3,0,1,1,0,7,0,13,0,7,0,1,1,0,15,0,83,0,220,0,303,0,220,

%T 0,83,0,15,0,1,1,0,31,0,413,0,3141,0,15261,0,50187,0,115410,0,189036,

%U 0,222621,0,189036,0,115410,0,50187,0,15261,0,3141,0,413,0,31,0,1,1,0,63

%N Coefficients of numerator polynomials P(n,x) associated with reciprocation.

%C 1. P(n,1)=A073833(n) for n>=1; P(n,2)=A073833(n+1) for n>=0.

%C 2. P(n)=P(n-1)^2+P(n-1)*P(n-2)^2-P(n-2)^4 for n>=3.

%C 3. For n>=3, P(n)=P(n,x)=S(n,i*x), where S(n) is the polynomial at A147985.

%C Thus all the zeros of P(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 P(1) = x

%e P(2) = x^2+1

%e P(3) = x^4+3*x^2+1

%e P(4) = x^8+7*x^6+13*x^4+7x^2+1

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

%e 1 0

%e 1 0 1

%e 1 0 3 0 1

%e 1 0 7 0 13 0 7 0 1

%t p[1] = x; q[1] = 1; p[n_] := p[n] = p[n-1]^2 + q[n-1]^2; q[n_] := q[n] = p[n-1]*q[n-1]; row[n_] := CoefficientList[p[n], x] // Reverse; Table[row[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Apr 22 2013 *)

%Y Cf. A147985, A147986, A147988, A147989, A147990, A147991, A147992, A147993.

%K nonn,tabf

%O 1,8

%A _Clark Kimberling_, Nov 24 2008