A280709 The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1.

1, 3, 6, 10, 16, 24, 38, 58, 86, 122, 172, 236, 328, 448, 606, 802, 1060, 1380, 1806, 2338, 3018, 3846, 4900, 6180, 7816, 9808, 12294, 15274, 18982, 23418, 28938, 35542, 43638, 53226, 64942, 78786, 95686, 115642, 139754, 168022, 202086, 241946

Offset: 0

Author

Christopher J. Smyth, Jan 07 2017

Keywords

• easy
• nonn

Comments

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

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.

Crossrefs

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

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

Programs

Mathematica

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 *)

Formula

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

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.

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

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