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 A051894 Number of monic polynomials with integer coefficients of degree n with all roots in unit disc. 4
 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; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. - Franklin T. Adams-Watters, Jul 19 2006 First differences give A120963. - Joerg Arndt, Nov 22 2014 REFERENCES Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, Technical Report TR\16\1999, University of Cyprus. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, American Math. Monthly, 108, 253-257 (2001). FORMULA 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. - Franklin T. Adams-Watters, Jul 19 2006 EXAMPLE 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. MATHEMATICA max = 40; CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 1, 5max}] + O[x]^max, x] // Accumulate (* Jean-François Alcover, Apr 14 2017 *) PROG (PARI) N=66; x='x+O('x^N); Ph(n)=if(n==0, 1, eulerphi(n)); Vec(1/prod(n=0, N, 1-x^Ph(n))) \\ Joerg Arndt, Jul 10 2015 CROSSREFS Cf. A014197, A120963. Sequence in context: A285927 A147371 A075188 * A146393 A147431 A329145 Adjacent sequences:  A051891 A051892 A051893 * A051895 A051896 A051897 KEYWORD nice,nonn AUTHOR Pantelis Damianou, Dec 17 1999 EXTENSIONS More terms from Franklin T. Adams-Watters, Jul 19 2006 STATUS approved

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Last modified September 24 21:35 EDT 2020. Contains 337322 sequences. (Running on oeis4.)