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%I #62 May 06 2024 12:31:03
%S 1,2,6,10,24,38,78,118,224,330,584,838,1420,2002,3258,4514,7134,9754,
%T 15010,20266,30532,40798,60280,79762,115966,152170,217962,283754,
%U 401250,518746,724866,930986,1287306,1643626,2250538,2857450,3878298,4899146,6594822
%N 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.
%C Also the number of types of crystallographic rotations and reflection-rotations in n-dimensional Euclidean space. - _Andrey Zabolotskiy_, Jul 08 2017
%H Alois P. Heinz, <a href="/A120963/b120963.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H Gaëtan Chenevier, <a href="https://arxiv.org/abs/2002.03707">The Characteristic Masses of Niemeier Lattices</a>, arXiv:2002.03707 [math.NT], 2020.
%H Peter Engel, Louis Michel and Marjorie Senechal, <a href="http://www.ihes.fr/~/vergne/LouisMichel/publications/LatticeGeometry.pdf">Lattice Geometry</a>, 2004 (see section 1.4.3).
%H Richard P. Stanley, <a href="https://math.mit.edu/~rstan/papers/cyclotomic.pdf">Some enumerative applications of cyclotomic polynomials</a>, MIT, 2024. See p. 15.
%H D. Weigel, R. Veysseyre, T. Phan, J. M. Effantin, and Y. Billiet, <a href="https://doi.org/10.1107/S0108767384000702">Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations</a>, Acta Cryst., A40 (1984), 323-330 (see table 3).
%F Euler transform of A014197.
%F G.f.: Product_{k>=1} 1/(1-x^phi(k)) = Product_{j>=1} (1-x^j)^(-A014197(j)). - _Christopher J. Smyth_, Jan 08 2017
%F log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - _Vaclav Kotesovec_, Sep 02 2021
%e 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.
%e The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.
%p with(numtheory):
%p b:= proc(n) option remember; nops(invphi(n)) end:
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p a(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jul 04 2019
%t terms = 40;
%t S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),
%t {k, 1, m*terms}] + O[x]^terms, x];
%t S[m = 1];
%t S[m++];
%t While[S[m] != S[m-1], m++];
%t S[m] (* _Jean-François Alcover_, Apr 14 2017, after _Christopher J. Smyth_, updated May 13 2022 *)
%Y Cf. A014197, A051894, A280611 (variant where repeated roots are not allowed).
%Y See also A341710, A341711, A341712.
%K easy,nonn
%O 0,2
%A _Franklin T. Adams-Watters_, Jul 19 2006
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