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

%I A013595 #75 Feb 16 2025 08:32:32
%S A013595 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,
%T A013595 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 A013595 1,1,1,1,1,1,1,1,-1,1,-1,1,-1,1,1,-1,0,1,-1,1,0,-1,1
%N A013595 Irregular triangle read by rows: coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order).
%C A013595 We follow Maple in defining Phi_0 to be x; it could equally well be taken to be 1.
%C A013595 From _Wolfdieter Lang_, Oct 29 2013: (Start)
%C A013595 The length of row n  >= 1 of this  table is phi(n) + 1 = A000010(n) + 1. Row n = 0 has here length 2.
%C A013595 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.
%C A013595 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.
%C A013595 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.
%C A013595 (End)
%C A013595 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
%D A013595 E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968; see p. 90.
%D A013595 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 325.
%D A013595 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1991, p. 137.
%D A013595 K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, p. 194.
%H A013595 Robert Israel, <a href="/A013595/b013595.txt">Table of n, a(n) for n = 0..10000</a>
%H A013595 Emma Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06309-3">On the magnitude of the coefficients of the cyclotomic polynomial</a>, Bull. Amer. Math. Soc. 42 (1936), 389-392.
%H A013595 Rakshith Rajashekar, Marco Di Renzo, K.V.S. Hari, L. Hanzo, <a href="https://eprints.soton.ac.uk/414186/1/Full_duplex_SM_Drones.pdf">A generalised transmit and receive diversity condition for feedback assisted MIMO systems: theory & applications in full-duplex spatial modulation</a>, 2017.
%H A013595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%H A013595 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic Polynomial</a>.
%F A013595 a(n,m) = [x^m] Phi_n(x), n >= 0, 0 <= m <= phi(n), with phi(n) = A000010(n). - _Wolfdieter Lang_, Oct 29 2013
%e A013595 Phi_0 = x; Phi_1 = x - 1; Phi_2 = x + 1; Phi_3 = x^2 + x + 1; Phi_4 = x^2 + 1; ...
%e A013595 From _Wolfdieter Lang_, Oct 29 2013: (Start)
%e A013595 The irregular triangle a(n,m) begins:
%e A013595 n\m 0  1  2  3  4  5  6  7  8  9 10 11 12 ...
%e A013595 0:  0  1
%e A013595 1: -1  1
%e A013595 2:  1  1
%e A013595 3:  1  1  1
%e A013595 4:  1  0  1
%e A013595 5:  1  1  1  1  1
%e A013595 6:  1 -1  1
%e A013595 7:  1  1  1  1  1  1  1
%e A013595 8:  1  0  0  0  1
%e A013595 9:  1  0  0  1  0  0  1
%e A013595 10: 1 -1  1 -1  1
%e A013595 11: 1  1  1  1  1  1  1  1  1  1  1
%e A013595 12: 1  0 -1  0  1
%e A013595 13: 1  1  1  1  1  1  1  1  1  1  1  1  1
%e A013595 14: 1 -1  1 -1  1 -1  1
%e A013595 15: 1 -1  0  1 -1  1  0 -1  1
%e A013595 ...
%e A013595 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.
%e A013595 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.
%e A013595 (End)
%p A013595 N:= 100:  # to get coefficients up to cyclotomic(N,x)
%p A013595 with(numtheory):
%p A013595 for n from 0 to N do
%p A013595   C:= cyclotomic(n,x);
%p A013595   L[n]:= seq(coeff(C,x,i),i=0..degree(C));
%p A013595 od:
%p A013595 A:= [seq](L[n],n=0..N): # note that A013595(n) = A[n+1]
%p A013595 # _Robert Israel_, Apr 17 2014
%t A013595 Table[CoefficientList[x^KroneckerDelta[n] Cyclotomic[n, x], x], {n, 0, 15}] // Flatten (* _Peter Luschny_, Dec 27 2016 *)
%o A013595 (PARI) row(n) = if (n==0, p=x, p = polcyclo(n)); Vecrev(p); \\ _Michel Marcus_, Dec 14 2015
%Y A013595 Cf. A013596, A020500 (row sums, n >= 1), A020513 (alternating row sums).
%Y A013595 For record coefficients see A160340, A262404, A262405, A278567.
%Y A013595 Column m=1 is A157657.
%K A013595 sign,easy,nice,tabf
%O A013595 0,3440
%A A013595 _N. J. A. Sloane_
%E A013595 Maple program corrected by _Robert Israel_, Apr 17 2014