login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Number of monic polynomials with integer coefficients of degree n with all roots in unit disc.
6

%I #30 Sep 02 2021 04:26:44

%S 1,3,9,19,43,81,159,277,501,831,1415,2253,3673,5675,8933,13447,20581,

%T 30335,45345,65611,96143,136941,197221,276983,392949,545119,763081,

%U 1046835,1448085,1966831,2691697,3622683,4909989,6553615,8804153

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

%C 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.

%C The irreducible polynomials with this property consist of f(x)=x plus the cyclotomic polynomials. - _Franklin T. Adams-Watters_, Jul 19 2006

%C First differences give A120963. - _Joerg Arndt_, Nov 22 2014

%D Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, Technical Report TR\16\1999, University of Cyprus.

%H Vaclav Kotesovec, <a href="/A051894/b051894.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H Pantelis A. Damianou, <a href="http://www.jstor.org/stable/2695387">Monic polynomials in Z[x] with roots in the unit disc</a>, American Math. Monthly, 108, 253-257 (2001).

%F 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

%F log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - _Vaclav Kotesovec_, Sep 02 2021

%e 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.

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

%o (PARI) N=66; x='x+O('x^N); Ph(n)=if(n==0,1,eulerphi(n));

%o Vec(1/prod(n=0,N,1-x^Ph(n))) \\ _Joerg Arndt_, Jul 10 2015

%Y Cf. A014197, A120963.

%K nice,nonn

%O 0,2

%A _Pantelis Damianou_, Dec 17 1999

%E More terms from _Franklin T. Adams-Watters_, Jul 19 2006