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

Author

Franklin T. Adams-Watters, Jul 19 2006

Keywords

• easy
• nonn

Comments

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

Examples

			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.
		

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).

Crossrefs

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

See also A341710, A341711, A341712.

Programs

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 *)

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