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A001782 Discriminants of Shapiro polynomials.
(Formerly M5286 N2301)
5

%I M5286 N2301

%S 1,-44,-4940800,-564083990621761115783168,

%T -265595429519150677725101890892978815884074732203939261150723571712

%N Discriminants of Shapiro polynomials.

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

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

%H Sean A. Irvine, <a href="/A001782/b001782.txt">Table of n, a(n) for n = 1..8</a>

%H Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.

%H John Brillhart and L. Carlitz, <a href="https://doi.org/10.1090/S0002-9939-1970-0260955-6">Note on the Shapiro polynomials</a>, Proceedings of the American Mathematical Society, volume 25, number 1, May 1970, pages 114-118. Also <a href="http://www.jstor.org/stable/2036537">at JSTOR</a>, or <a href="/A001782/a001782.pdf">annotated scanned copy</a>.

%H Robert Davis, Greg Simay, <a href="https://arxiv.org/abs/2001.11089">Further Combinatorics and Applications of Two-Toned Tilings</a>, arXiv:2001.11089 [math.CO], 2020.

%F 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

%o (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

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

%K sign,nice

%O 1,2

%A _N. J. A. Sloane_.

%E Extended by _Sean A. Irvine_, Nov 25 2012

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Last modified June 12 10:29 EDT 2021. Contains 344946 sequences. (Running on oeis4.)