A306453 Triangle of coefficients of "inverse" cyclotomic polynomial Psi_n(x) (exponents in increasing order).

0, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, 0, 0, 0, 1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 1, 1, -1, 1, -1, 0, -1, 0, 0, 0, 1, 0, 1, -1, 1, -1, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, 0, 0, 1, 1, 1

Offset

0

Comments

The first polynomial showing a coefficient other than -1 or 1 is Psi_561(x). Curiously, 561 is the smallest Carmichael number.

Links

Table of n, a(n) for n=0..63.

Pieter Moree, Inverse cyclotomic polynomials, Journal of Number Theory, Volume 129, Issue 3, March 2009, Pages 667-680.

Eric Weisstein's MathWorld, Cyclotomic Polynomial.

Wikipedia, Cyclotomic Polynomial.

Index to sequences related to inverse of cyclotomic polynomials

Formula

Phi_n(x) * Psi_n(x) = x^n - 1, where Phi_n(x) is the n-th cyclotomic polynomial.

Example

Phi_10(x)*Psi_10(x) = (1-x+x^2-x^3+x^4)*(-1-x+x^5+x^6) = -1+x^10.
Inverse cyclotomic polynomials begin:
n:   Psi_n(x)
0:   0,
1:   1,
2:  -1 + x,
3:  -1 + x,
4:  -1 + x^2,
5:  -1 + x,
6:  -1 - x + x^3 + x^4,
7:  -1 + x,
8:  -1 + x^4,
9:  -1 + x^3,
10: -1 - x + x^5 + x^6
...
Coefficients begin:
0:   0;
1:   1;
2:  -1,  1;
3:  -1,  1;
4:  -1,  0,  1;
5:  -1,  1;
6:  -1, -1,  0,  1, 1;
7:  -1,  1;
8:  -1,  0,  0,  0,  1;
9:  -1,  0,  0,  1;
10: -1, -1,  0,  0,  0,  1,  1;
...

Mathematica

Psi[n_, x_] := PolynomialQuotient[x^n-1, Cyclotomic[n, x], x]; Psi[0, _]=0;
row[n_] := CoefficientList[Psi[n, x], x]; row[0] = {0};
Table[row[n], {n, 0, 15}] // Flatten

Crossrefs

Cf. A013595, A131672, A180093, A291137, A333248.

Row lengths are A062830.

Sequence in context: A211487 A101040 A341591 * A175629 A109720 A022932

Adjacent sequences: A306450 A306451 A306452 * A306454 A306455 A306456

Keyword

sign,tabf

Author

Jean-François Alcover, Feb 16 2019

Status

approved