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

Author

Pantelis Damianou, Dec 17 1999

Keywords

• nice
• nonn

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

Examples

			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.
		

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).

Crossrefs

Cf. A014197, A120963.

Programs

Mathematica

max = 40; CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 1, 5max}] + O[x]^max, x] // Accumulate (* Jean-François Alcover, Apr 14 2017 *)

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

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

Extensions

More terms from Franklin T. Adams-Watters, Jul 19 2006