A001782 Discriminants of Shapiro polynomials. (Formerly M5286 N2301)

1, -44, -4940800, -564083990621761115783168, -265595429519150677725101890892978815884074732203939261150723571712

Offset

1,2

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Sean A. Irvine, Table of n, a(n) for n = 1..8

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.

John Brillhart and L. Carlitz, Note on the Shapiro polynomials, Proceedings of the American Mathematical Society, volume 25, number 1, May 1970, pages 114-118.  Also at JSTOR, or annotated scanned copy.

Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.

Formula

Let P_0(x) = Q_0(x) = 1. For n > 0, P_{n + 1}(x) = P_n(x) + x^(2^n)*Q_n(x) and Q_{n + 1}(x) = P_n(x) - x^(2^n)*Q_n(x). Then, a(n) = discrim(P_n(x)). Note also that discrim(P_n(x)) = discrim(Q_n(x)). - Sean A. Irvine, Nov 25 2012

Prog

(PARI) a(n) = my(P=Pol(1), Q=1); for(i=0, n-1, [P, Q]=[P+'x^(2^i)*Q, P-'x^(2^i)*Q]); poldisc(P); \\ Kevin Ryde, Feb 23 2020

Crossrefs

See A020985 for the Shapiro polynomials. Cf. A331691 (P,Q resultant).

Sequence in context: A115734 A119078 A172878 * A172910 A119058 A218402

Adjacent sequences: A001779 A001780 A001781 * A001783 A001784 A001785

Keyword

sign,nice

Author

N. J. A. Sloane.

Extensions

Extended by Sean A. Irvine, Nov 25 2012

Status

approved