A076585 Let P(n,x) = Product_{k=1..n} polcyclo(k,x) where polcyclo(k,x) denotes the k-th cyclotomic polynomial. Sequence gives array of coefficients of P(n,x).

1, 1, -1, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, 1, 0, -1, -1, -1, 1, 2, 3, 3, 2, 0, -2, -3, -3, -2, -1, 1, 1, 2, 2, 2, 1, 0, -1, -2, -2, -2, -1, -1, 1, 2, 4, 6, 8, 9, 9, 7, 4, 0, -4, -7, -9, -9, -8, -6, -4, -2, -1, 1, 2, 4, 6, 9, 11, 13, 13, 12, 9, 5, 0, -5, -9, -12, -13, -13, -11, -9, -6, -4, -2, -1, 1, 2, 4, 7, 11, 15, 20, 24, 27, 28, 27, 23, 17, 9, 0, -9

Offset

0,20

Comments

The degree of P(n,x) is phi(1) + phi(2) + ... + phi(n) = A002088(n) and if c(n,i) denotes the coefficient of x^i in P(n,x): c(n,i) + c(n, A002088(n) -i ) = 0.

Links

Alois P. Heinz, Rows n = 0..60, flattened

Example

P(5,x) = x^10 + 2*x^9 + 3*x^8 + 3*x^7 + 2*x^6 - 2*x^4 - 3*x^3 - 3*x^2 - 2*x - 1 hence: 1,2,3,3,2,0,-2,-3,-3,-2,-1 is a segment in the sequence.
Triangle begins:
  [1]
  [1, -1]
  [1,  0, -1]
  [1,  1,  0, -1, -1]
  [1,  1,  1,  0, -1, -1, -1]
  [1,  2,  3,  3,  2,  0, -2, -3, -3, -2, -1]
  [1,  1,  2,  2,  2,  1,  0, -1, -2, -2, -2, -1, -1]
  [1,  2,  4,  6,  8,  9,  9,  7,  4,  0, -4, -7, -9, -9, -8, -6, -4, -2, -1]
  ...

Maple

T:= n-> (p-> seq(coeff(p, x, degree(p)-i), i=0..degree(p)))(
             mul(numtheory[cyclotomic](i, x), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 15 2022

Mathematica

P[n_, x_] := Product[Cyclotomic[k, x], {k, 1, n}];
T[n_] := CoefficientList[P[n, x], x] // Reverse;
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 07 2025 *)

Prog

(PARI) row(n) = Vec(prod(k=1, n, polcyclo(k, x))); \\ Michel Marcus, May 24 2019

Crossrefs

Cf. A002088, A013595, A013596.

Row sums give A000007.

Sequence in context: A198197 A203400 A077869 * A323186 A022906 A285408

Adjacent sequences: A076582 A076583 A076584 * A076586 A076587 A076588

Keyword

sign,look,tabf,changed

Author

Benoit Cloitre, Oct 20 2002

Extensions

Keyword tabf from Michel Marcus, May 24 2019

Status

approved