A154305 Coefficients of polynomials H(n,x) associated with squares of polynomials S(n,x).

1, 0, 1, 1, 0, -6, 0, 1, 1, 0, 20, 0, -26, 0, 20, 0, 1, 1, 0, -88, 0, 92, 0, -872, 0, 1990, 0, -872, 0, 92, 0, -88, 0, 1, 1, 0, 336, 0, -3336, 0, 6961, 0, -77796, 0, -647088, 0, 2618568, 0, -3600784, 0, 3346502, 0, -3600784, 0, 2618568, 0, -647088, 0, -77796, 0

Offset: 1

Author

Clark Kimberling, Jan 06 2009

Keywords

• sign
• tabf

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).

Examples

			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.
		

Links

• Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Crossrefs

Cf. A147985, A147986.

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]