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A147986
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Coefficients of denominator polynomials T(n,x) associated with reciprocation.
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7
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1, 1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 4, 0, -1, 0, 1, 0, -11, 0, 45, 0, -88, 0, 88, 0, -45, 0, 11, 0, -1, 0, 1, 0, -26, 0, 293, 0, -1896, 0, 7866, 0, -22122, 0, 43488, 0, -60753, 0, 60753, 0, -43488, 0, 22122, 0, -7866, 0, 1896, 0, -293, 0, 26, 0, -1, 0, 1, 0, -57, 0, 1512, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,10
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COMMENTS
| T(n)=S(1)*S(2)*...*S(n-1). The degree of S(n) in x is m=2^(n-1), so that the degree of T(n) is m-1. Write the zeros of T(n) as r(1)<r(2)<...<r(m-1) and the zeros of S(n) as z(1)<z(2)<...<z(m). Then z(1)<r(1)<z(2)<r(2)<...<r(m-1)<z(m); i.e., the zeros of T(n) intersperse the zeros of S(n).
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REFERENCES
| Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.
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FORMULA
| The basic idea is to iterate the reciprocation-difference mapping
x/y -> x/y-y/x. Let x be an indeterminate, S(1)=x, T(1)=1 and for n>1,
define S(n)=S(n-1)^2-T(n-1)^2 and T(n)=S(n-1)*T(n-1), so that
S(n)/T(n)=S(n-1)/T(n-1)-T(n-1)/S(n-1).
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EXAMPLE
| T(1)=1
T(2)=x
T(3)=x^3-x
T(4)=x^7-4*x^5+4*x^3-x
so that as an array A147986 begins with
1
1 0
1 0 -1 0
1 0 -4 0 4 0 -1 0
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CROSSREFS
| Cf. A147985, A147987, A147988, A147989, A147990, A147991, A147992, A147993.
Sequence in context: A028634 A170773 A028618 * A147988 A019920 A010675
Adjacent sequences: A147983 A147984 A147985 * A147987 A147988 A147989
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KEYWORD
| sign,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Nov 24 2008
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