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 A013595 Triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order). 24
 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 (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_{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 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. 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 Table[CoefficientList[x^KroneckerDelta[n] Cyclotomic[n, x], x], {n, 0, 15}] // Flatten (* Peter Luschny, Dec 27 2016 *) 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). For record coefficients see A160340, A262404, A262405, A278567. Sequence in context: A188642 A168046 A168184 * A011582 A145568 A168185 Adjacent sequences:  A013592 A013593 A013594 * A013596 A013597 A013598 KEYWORD sign,easy,nice,tabf AUTHOR EXTENSIONS Maple program corrected by Robert Israel, Apr 17 2014 STATUS approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)