A280709 - OEIS

%I #21 Sep 02 2021 17:28:06

%S 1,3,6,10,16,24,38,58,86,122,172,236,328,448,606,802,1060,1380,1806,

%T 2338,3018,3846,4900,6180,7816,9808,12294,15274,18982,23418,28938,

%U 35542,43638,53226,64942,78786,95686,115642,139754,168022,202086,241946

%N The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1.

%C Such polynomials are a product of distinct cyclotomic polynomials, possibly multiplied by z. This follows from a classical result of Kronecker -- see Links.

%H Vaclav Kotesovec, <a href="/A280709/b280709.txt">Table of n, a(n) for n = 0..10000</a>

%H L. Kronecker, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002149613">Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten</a>, J. Reine Angew. Math. 53 (1857), 173-175.

%F a(0) = 1 and a(n) = b(n)+b(n-1) for n >= 1, where b(n) = A280611(n).

%F G.f.: (1+x)*Product_{i>=1} (1+x^phi(i)) = (1+x)*Product_{j>=1} (1+x^j)^A014197(j), where phi(i)=A000010(i) is Euler's totient function.

%F It is also the Euler transform of A280712 except with its first two terms (2,1) replaced by (3,0).

%F a(n) ~ exp(sqrt(105*zeta(3)*n/2)/Pi) * (105*zeta(3)/2)^(1/4) / (2*Pi*n^(3/4)). - _Vaclav Kotesovec_, Sep 02 2021

%e a(2)=6 because the six polynomials z^2+z+1, z^2+1, z^2-z+1, z^2-z, z^2+z and z^2-1 are the only ones of the required type.

%t Table[SeriesCoefficient[(1 + x) Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 120}] (* _Michael De Vlieger_, Jan 10 2017 *)

%Y Cf. A280611 (variant where all roots must have modulus exactly 1);

%Y Cf. A120963 (variant where multiple roots are allowed).

%K easy,nonn

%O 0,2

%A _Christopher J. Smyth_, Jan 07 2017