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Search: id:A147988
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| A147988 |
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Coefficients of denominator polynomials Q(n,x) associated with reciprocation. |
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+0
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; listen)
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OFFSET
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1,10
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COMMENT
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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
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Clark Kimberling, "Polynomials associated with reciprocation," preprint, 2008.
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FORMULA
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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
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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
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Cf. A147985, A147986, A147987, A147989, A147990, A147991, A147992, A147993.
Sequence in context: A028634 A028618 A147986 this_sequence A019920 A010675 A035673
Adjacent sequences: A147985 A147986 A147987 this_sequence A147989 A147990 A147991
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Nov 24 2008
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