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Search: id:A051894
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| A051894 |
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Number of monic polynomials with integer coefficients of degree n with all roots in unit disc. |
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+0
2
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| 1, 3, 9, 19, 43, 81, 159, 277, 501, 831, 1415, 2253, 3673, 5675, 8933, 13447, 20581, 30335, 45345, 65611, 96143, 136941, 197221, 276983, 392949, 545119, 763081, 1046835, 1448085, 1966831, 2691697, 3622683, 4909989, 6553615, 8804153
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The number of polynomials of a given degree that satisfy the conditions {1) monic, 2) integer coefficients and 3) all roots in the unit disc} is finite. This is an old theorem of Kronecker.
The irreducible polynomials with this property consist of f(x)=x plus the cyclotomic polynomials. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 19 2006
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REFERENCES
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Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, Technical Report TR\16\1999, University of Cyprus.
Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, American Math. Monthly, 108, 253-257 (2001)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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Euler transform of b(n) where b(n) = A014197(n) except for n=1, where b(n) = 3 instead of 2; cumulative sum of A120963. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 19 2006
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EXAMPLE
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a(1)=3 because the only monic, linear, polynomials with coefficients in Z and all their roots in the unit disc are f(z)=z, g(z)=z-1, h(z)=z+1
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CROSSREFS
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Cf. A014197, A120963.
Sequence in context: A153084 A147371 A075188 this_sequence A146393 A147431 A147334
Adjacent sequences: A051891 A051892 A051893 this_sequence A051895 A051896 A051897
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KEYWORD
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nice,nonn
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AUTHOR
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Pantelis A. Damianou (damianou(AT)ucy.ac.cy), Dec 17 1999
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EXTENSIONS
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More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 19 2006
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