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

%I A051894 #30 Sep 02 2021 04:26:44
%S A051894 1,3,9,19,43,81,159,277,501,831,1415,2253,3673,5675,8933,13447,20581,
%T A051894 30335,45345,65611,96143,136941,197221,276983,392949,545119,763081,
%U A051894 1046835,1448085,1966831,2691697,3622683,4909989,6553615,8804153
%N A051894 Number of monic polynomials with integer coefficients of degree n with all roots in unit disc.
%C A051894 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 A051894 The irreducible polynomials with this property consist of f(x)=x plus the cyclotomic polynomials. - _Franklin T. Adams-Watters_, Jul 19 2006
%C A051894 First differences give A120963. - _Joerg Arndt_, Nov 22 2014
%D A051894 Pantelis A. Damianou, Monic polynomials in Z[x] with roots in the unit disc, Technical Report TR\16\1999, University of Cyprus.
%H A051894 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 A051894 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 A051894 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 A051894 log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - _Vaclav Kotesovec_, Sep 02 2021
%e A051894 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 A051894 max = 40; CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 1, 5max}] + O[x]^max, x] // Accumulate (* _Jean-François Alcover_, Apr 14 2017 *)
%o A051894 (PARI) N=66; x='x+O('x^N); Ph(n)=if(n==0,1,eulerphi(n));
%o A051894 Vec(1/prod(n=0,N,1-x^Ph(n))) \\ _Joerg Arndt_, Jul 10 2015
%Y A051894 Cf. A014197, A120963.
%K A051894 nice,nonn
%O A051894 0,2
%A A051894 _Pantelis Damianou_, Dec 17 1999
%E A051894 More terms from _Franklin T. Adams-Watters_, Jul 19 2006