A013596 Irregular triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in decreasing order).

1, 0, 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

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.

Examples

			Phi_0 = x             --> Row 0: [1, 0]
Phi_1 = x - 1         --> Row 1: [1, -1]
Phi_2 = x + 1         --> Row 2: [1, 1]
Phi_3 = x^2 + x + 1   --> Row 3: [1, 1, 1]
Phi_4 = x^2 + 1       --> Row 4: [1, 0, 1]
etc. After row zero, each row n has A039649(n) terms.
		

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

Links

• Antti Karttunen, Table of n, a(n) for n = 0..45566, rows 0..385 flattened

Crossrefs

Version with reversed rows: A013595.

Cf. A039649, A160340.

Programs

Maple

with(numtheory): [ seq(cyclotomic(n,x), n=0..48) ];

Mathematica

Join[{1, 0}, Table[ CoefficientList[ Cyclotomic[n, x], x] // Reverse, {n, 1, 16}] // Flatten] (* Jean-François Alcover, Dec 11 2012 *)

PARI

A013595row(n) = { if(!n, p=x, p = polcyclo(n)); Vecrev(p); }; \\ This function from Michel Marcus's code for A013595.
n=0; for(r=0,385,v=A013595row(r);k=length(v);while(k>0,write("b013596.txt", n, " ", v[k]);n=n+1;k=k-1)); \\ Antti Karttunen, Aug 13 2017

Extensions

Example section edited by Antti Karttunen, Aug 13 2017