A051894 Number of monic polynomials with integer coefficients of degree n with all roots in unit disc.

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

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

log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - Vaclav Kotesovec, Sep 02 2021

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