A086783 Discriminant of the polynomial x^n - 1.

1, 4, -27, -256, 3125, 46656, -823543, -16777216, 387420489, 10000000000, -285311670611, -8916100448256, 302875106592253, 11112006825558016, -437893890380859375, -18446744073709551616, 827240261886336764177, 39346408075296537575424, -1978419655660313589123979

Offset

1,2

Comments

By definition, a(n) = Product_{1<=i<j<=n} (r^i - r^j)^2, where r = exp(2*Pi*i/n). As a result we have a(n) = det(M)^2, where M is the n X n matrix M_{jk} = r^(s(j)*t(k)), defined for any permutations {s(1), s(2), ..., s(n)}, {t(1), t(2), ..., t(n)} of {1,2,...n}. - Jianing Song, Mar 17 2021

Links

Kenneth G. Hawes, Table of n, a(n) for n = 1..386

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 249-255. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Formula

a(n) = (-1)^floor((n-1)/2) * n^n = (-1)^floor((n-1)/2) * A000312(n).

Prog

(Sage) def A086783(n) : return (-1)^((n-1)//2) * n^n # Eric M. Schmidt, May 04 2013
(PARI) a(n)=poldisc('x^n-1); \\ Joerg Arndt, May 04 2013

Crossrefs

Cf. A000312, A004124.

Sequence in context: A245414 A177885 A000312 * A301742 A050764 A302108

Adjacent sequences: A086780 A086781 A086782 * A086784 A086785 A086786

Keyword

sign,easy

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

Extensions

More terms from Eric M. Schmidt, May 04 2013

Status

approved