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

%I A154305 #12 Jan 23 2019 10:20:34
%S A154305 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,
%T A154305 -872,0,92,0,-88,0,1,1,0,336,0,-3336,0,6961,0,-77796,0,-647088,0,
%U A154305 2618568,0,-3600784,0,3346502,0,-3600784,0,2618568,0,-647088,0,-77796,0
%N A154305 Coefficients of polynomials H(n,x) associated with squares of polynomials S(n,x).
%C A154305 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).
%H A154305 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 A154305 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).
%F A154305 H(n,x)=2*H(n-2,x)^4-H(n-1,x)^2. [_Clark Kimberling_, Mar 19 2009]
%e A154305 H(3,x)=x^2+1 and S(3,x)=(x^2+1)^2.
%e A154305 H(4,x)=x^4-6*x^2+1
%e A154305 H(6,x)=x^8+20*x^6-26*x^4+20*x^2+1.
%e A154305 First three rows:
%e A154305 1 0 1
%e A154305 1 0 -6 0 1
%e A154305 1 0 20 0 -26 0 20 0 1.
%Y A154305 Cf. A147985, A147986.
%K A154305 sign,tabf
%O A154305 1,6
%A A154305 _Clark Kimberling_, Jan 06 2009