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A280709
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The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1.
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3
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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
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OFFSET
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0,2
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COMMENTS
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Such polynomials are a product of distinct cyclotomic polynomials, possibly multiplied by z. This follows from a classical result of Kronecker -- see Links.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A280611 (variant where all roots must have modulus exactly 1);
Cf. A120963 (variant where multiple roots are allowed).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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