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

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

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, MIT, 2024. See p. 15.

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).

See also A341710, A341711, A341712.

Sequence in context: A072297 A358908 A183036 * A370587 A200220 A188224

Adjacent sequences: A120960 A120961 A120962 * A120964 A120965 A120966

Keyword

easy,nonn

Author

Franklin T. Adams-Watters, Jul 19 2006

Status

approved