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A147988
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Coefficients of denominator polynomials Q(n,x) associated with reciprocation.
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6
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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, 24858, 0, 284578, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,10
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COMMENTS
| 1. Q(n,1)=A073834(n) for n>=1.
2. For n>=3, Q(n)=Q(n,x)=i*T(n,i*x), where T(n) is the polynomial at A147986.
Thus all the zeros of Q(n,x), for n>=2, are nonreal.
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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-sum mapping
x/y -> x/y+y/x. Let x be an indeterminate, P(1)=x, Q(1)=1 and for n>1,
define P(n)=P(n-1)^2+Q(n-1)^2 and Q(n)=P(n-1)*Q(n-1), so that
P(n)/Q(n)=P(n-1)/Q(n-1)-Q(n-1)/P(n-1).
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EXAMPLE
| Q(1)=1
Q(2)=x
Q(3)=x^3+x
Q(4)=x^7+4*x^5+4*x^3+1
so that as an array A147988 begins with
1
1 0
1 0 1 0
1 0 4 0 4 0 1
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CROSSREFS
| Cf. A147985, A147986, A147987, A147989, A147990, A147991, A147992, A147993.
Sequence in context: A170773 A028618 A147986 * A019920 A010675 A035673
Adjacent sequences: A147985 A147986 A147987 * A147989 A147990 A147991
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Nov 24 2008
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