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A154305
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Coefficients of polynomials H(n,x) associated with squares of polynomials S(n,x).
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0
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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
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OFFSET
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1,6
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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