A013595 Irregular triangle read by rows: coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• tabf

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_{k=0..n-1, gcd(k,n)=1} (x - (omega_n)^k). 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_{rad(n)}(x^(n/rad(n))), n >= 2, with rad(n) = A007947(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

Examples

			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 irregular triangle 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)
		

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.
• Rakshith Rajashekar, Marco Di Renzo, K.V.S. Hari, L. Hanzo, A generalised transmit and receive diversity condition for feedback assisted MIMO systems: theory & applications in full-duplex spatial modulation, 2017.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.
• Wikipedia, Cyclotomic Polynomial.

Crossrefs

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

For record coefficients see A160340, A262404, A262405, A278567.

Column m=1 is A157657.

Programs

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

Table[CoefficientList[x^KroneckerDelta[n] Cyclotomic[n, x], x], {n, 0, 15}] // Flatten (* Peter Luschny, Dec 27 2016 *)

PARI

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

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

Extensions

Maple program corrected by Robert Israel, Apr 17 2014