A120963 - OEIS

A120963

Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

1, 2, 6, 10, 24, 38, 78, 118, 224, 330, 584, 838, 1420, 2002, 3258, 4514, 7134, 9754, 15010, 20266, 30532, 40798, 60280, 79762, 115966, 152170, 217962, 283754, 401250, 518746, 724866, 930986, 1287306, 1643626, 2250538, 2857450, 3878298, 4899146, 6594822

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Also the number of types of crystallographic rotations and reflection-rotations in n-dimensional Euclidean space. - Andrey Zabolotskiy, Jul 08 2017

REFERENCES

Boyd, David W.(3-BC); Montgomery, Hugh L.(1-MI), Cyclotomic partitions. In Number theory (Banff, AB, 1988), 7-25. Walter de Gruyter & Co., Berlin, 1990 ISBN:3-11-011723-1, MR1106647. [Asymptotics]

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

Gaëtan Chenevier, The Characteristic Masses of Niemeier Lattices, arXiv:2002.03707 [math.NT], 2020.

Peter Engel, Louis Michel and Marjorie Senechal, Lattice Geometry, 2004 (see section 1.4.3).

Richard P. Stanley, Some enumerative applications of cyclotomic polynomials, preprint, 2024-2025. See p. 13.

D. Weigel, R. Veysseyre, T. Phan, J. M. Effantin, and Y. Billiet, Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations, Acta Cryst., A40 (1984), 323-330 (see table 3).

FORMULA

Euler transform of A014197.

G.f.: Product_{k>=1} 1/(1-x^phi(k)) = Product_{j>=1} (1-x^j)^(-A014197(j)). - Christopher J. Smyth, Jan 08 2017

log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - Vaclav Kotesovec, Sep 02 2021

EXAMPLE

The six polynomials of degree 2 consist of 3 irreducible cyclotomic polynomials: x^2+1, x^2+x+1 and x^2-x+1 and 3 products of 2 linear cyclotomic polynomials: x^2+2x+1, x^2-1 and x^2-2x+1.

The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.

MAPLE

with(numtheory):

b:= proc(n) option remember; nops(invphi(n)) end:

a:= proc(n) option remember; `if`(n=0, 1, add(

a(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)

end:

seq(a(n), n=0..40); # Alois P. Heinz, Jul 04 2019

MATHEMATICA

terms = 40;

S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),

{k, 1, m*terms}] + O[x]^terms, x];

S[m = 1];

S[m++];

While[S[m] != S[m-1], m++];

S[m] (* Jean-François Alcover, Apr 14 2017, after Christopher J. Smyth, updated May 13 2022 *)

CROSSREFS

Cf. A014197, A051894, A280611 (variant where repeated roots are not allowed).