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A013595 Triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order). 10
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((x^d - 1)^(mu(n/d)), d divides n) 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 sqfkn(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)

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

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 *)

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

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Last modified November 21 12:45 EST 2014. Contains 249778 sequences.