OFFSET
1,6
COMMENTS
Define S(1)=S(1,x)=x and T(1)=T(1,x)=1; for n>=1, define S(n+1)=[S(n)]^2-[T(n)]^2 and T(n+1)=c*S(n)*T(n). The sole value of c for which S(n) is the square of a polynomial for all n>=3 is 2i, and [H(n,x)]^2 = S(n,x).
LINKS
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.
FORMULA
H(3,x)=x^2+1 and H(n+1,x)=[(2*i*x)^p]*H(n,i/(2*x)-ix/2) for n>=3, where p=2^n-2 and i=sqrt(-1).
H(n,x)=2*H(n-2,x)^4-H(n-1,x)^2. [Clark Kimberling, Mar 19 2009]
EXAMPLE
H(3,x)=x^2+1 and S(3,x)=(x^2+1)^2.
H(4,x)=x^4-6*x^2+1
H(6,x)=x^8+20*x^6-26*x^4+20*x^2+1.
First three rows:
1 0 1
1 0 -6 0 1
1 0 20 0 -26 0 20 0 1.
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Clark Kimberling, Jan 06 2009
STATUS
approved