login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A013595 Triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order). 14
0, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 0, 1, -1, 1, 0, -1, 1, 1, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

We follow Maple in defining Phi_0 to be x; it could equally well be taken to be 1.

From Wolfdieter Lang, Oct 29 2013: (Start)

The length of row n  >= 1 of this  table is phi(n) + 1 = A000010(n) + 1. Row n = 0 has here length 2.

Phi_n(x) is the minimal polynomial of omega_n := exp(i*2*Pi/n) over the rationals. Namely, Phi_n(x) = product(x - (omega_n)^k, k = 0..n-1, restricted by gcd(k,n) = 1). See the Graham et al. reference, 4.50 a, pp. 149, 506.

Phi_n(x) = Product_{d|n}(x^d - 1)^(mu(n/d)) with the Moebius function mu(n) = A008683(n), n >= 1. See the Graham et al. reference, 4.50 b, pp. 149, 506.

Phi_n(x) = Phi_{sqfk(n)}(x^(n/sqfk(n))), n>=2, with sqfk(n) = AA007947(n), the squarefree kernel of n. Proof from the preceding formula, where only squarefree n/d (A005117) from the set of divisors of n enter, by mapping each factor (numerator or denominator) of the left hand side to one of the right hand side and vice versa.

(End)

Each row can be considered as the last column of the companion matrix of the cyclotomic polynomial: A000010(n) is the size of such a square matrix, last column has opposite signs and the last term (before last term of each row in A013595) equal to A008683(n). - Eric Desbiaux, Dec 14 2015

REFERENCES

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968; see p. 90.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 325.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1991, p. 137.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, p. 194.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 42 (1936), 389-392.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.

Wikipedia, Cyclotomic Polynomial.

FORMULA

a(n,m) = [x^m]Phi_n(x), n >= 0, 0 <= m <= phi(n), with phi(n) = A000010(n). - Wolfdieter Lang, Oct 29 2013

EXAMPLE

Phi_0 = x; Phi_1 = x-1; Phi_2 = x+1; Phi_3 = x^2+x+1; Phi_4 = x^2+1; ...

From Wolfdieter Lang, Oct 29 2013: (Start)

The table a(n,m) begins:

n\m 0  1  2  3  4  5  6  7  8  9 10 11 12 ...

0:  0  1

1: -1  1

2:  1  1

3:  1  1  1

4:  1  0  1

5:  1  1  1  1  1

6:  1 -1  1

7:  1  1  1  1  1  1  1

8:  1  0  0  0  1

9:  1  0  0  1  0  0  1

10: 1 -1  1 -1  1

11: 1  1  1  1  1  1  1  1  1  1  1

12: 1  0 -1  0  1

13: 1  1  1  1  1  1  1  1  1  1  1  1  1

14: 1 -1  1 -1  1 -1  1

15: 1 -1  0  1 -1  1  0 -1  1

...

Phi_15(x) = (x^1 - 1)*((x^3 - 1)^(-1))*((x^5 - 1)^(-1))*(x^15 -1) because mu(15) = mu(1) = + 1 and mu(3) = mu(5) = -1. Hence Phi_15(x) = 1 - x + x^3 - x^4 + x^5 - x^7 + x^8, giving row n = 15.

Example for the reduction via the squarefree kernel: Phi_{12}(x) = Phi_{6}(x^(12/6)) = Phi_{6}(x^2). By the formula with the Mobius function Phi_6(x) = Phi_2(x^3)/Phi_2(x) = 1 - x + x^2 and with x -> x^2 this becomes Phi_{12}(x) = 1 - x^2 + x^4.

(End)

MAPLE

N:= 100:  # to get coefficients up to cyclotomic(N, x)

with(numtheory):

for n from 0 to N do

  C:= cyclotomic(n, x);

  L[n]:= seq(coeff(C, x, i), i=0..degree(C));

od:

A:= [seq](L[n], n=0..N): # note that A013595(n) = A[n+1]

# Robert Israel, Apr 17 2014

MATHEMATICA

lst={}; Do[lst=Join[lst, CoefficientList[Cyclotomic[n, x], x]], {n, 0, 20}]; lst (* T. D. Noe, Dec 06 2005 *)

PROG

(PARI) row(n) = if (n==0, p=x, p = polcyclo(n)); Vecrev(p); \\ Michel Marcus, Dec 14 2015

CROSSREFS

Cf. A013596, A020500 (row sums, n >= 1), A020513 (alternating row sums).

Sequence in context: A168182 A168046 A168184 * A011582 A145568 A123927

Adjacent sequences:  A013592 A013593 A013594 * A013596 A013597 A013598

KEYWORD

sign,easy,nice,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

Maple program corrected by Robert Israel, Apr 17 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 29 17:51 EDT 2016. Contains 275947 sequences.