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A001782 Discriminants of Shapiro polynomials.

%I A001782 M5286 N2301 #44 Jul 08 2025 16:27:03
%S A001782 1,-44,-4940800,-564083990621761115783168,
%T A001782 -265595429519150677725101890892978815884074732203939261150723571712
%N A001782 Discriminants of Shapiro polynomials.
%D A001782 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001782 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001782 Sean A. Irvine, <a href="/A001782/b001782.txt">Table of n, a(n) for n = 1..8</a>
%H A001782 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 A001782 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 A001782 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 A001782 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 A001782 (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 A001782 See A020985 for the Shapiro polynomials.  Cf. A331691 (P,Q resultant).
%K A001782 sign,nice
%O A001782 1,2
%A A001782 _N. J. A. Sloane_
%E A001782 Extended by _Sean A. Irvine_, Nov 25 2012