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

0,2

Comments

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

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.

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

Example

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.

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 *)

Crossrefs

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

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

Sequence in context: A078663 A173691 A376708 * A025222 A011902 A025004

Adjacent sequences: A280706 A280707 A280708 * A280710 A280711 A280712

Keyword

easy,nonn

Author

Christopher J. Smyth, Jan 07 2017

Status

approved