A001782 Discriminants of Shapiro polynomials.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• nice

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.

Crossrefs

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

Programs

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

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

Extensions

Extended by Sean A. Irvine, Nov 25 2012