A013596 - OEIS

%I #34 Jan 28 2023 12:20:49

%S 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,

%T 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,

%U 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

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

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

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

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

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

%H Antti Karttunen, <a href="/A013596/b013596.txt">Table of n, a(n) for n = 0..45566, rows 0..385 flattened</a>

%e Phi_0 = x             --> Row 0: [1, 0]

%e Phi_1 = x - 1         --> Row 1: [1, -1]

%e Phi_2 = x + 1         --> Row 2: [1, 1]

%e Phi_3 = x^2 + x + 1   --> Row 3: [1, 1, 1]

%e Phi_4 = x^2 + 1       --> Row 4: [1, 0, 1]

%e etc. After row zero, each row n has A039649(n) terms.

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

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

%o (PARI)

%o A013595row(n) = { if(!n, p=x, p = polcyclo(n)); Vecrev(p); }; \\ This function from _Michel Marcus_'s code for A013595.

%o 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

%Y Version with reversed rows: A013595.

%Y Cf. A039649, A160340.

%K sign,easy,nice,tabf

%O 0,3440

%A _N. J. A. Sloane_

%E Example section edited by _Antti Karttunen_, Aug 13 2017