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
KEYWORD
nice,nonn
AUTHOR
Pantelis Damianou, Dec 17 1999
EXTENSIONS
More terms from Franklin T. Adams-Watters, Jul 19 2006
STATUS
approved