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A127670 Discriminants of Chebyshev S-polynomials A049310.

%I A127670 #114 Apr 08 2025 03:33:47
%S A127670 1,4,32,400,6912,153664,4194304,136048896,5120000000,219503494144,
%T A127670 10567230160896,564668382613504,33174037869887488,2125764000000000000,
%U A127670 147573952589676412928,11034809241396899282944,884295678882933431599104,75613185918270483380568064
%N A127670 Discriminants of Chebyshev S-polynomials A049310.
%C A127670 a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
%C A127670 From _Rigoberto Florez_, Sep 02 2018: (Start)
%C A127670 a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
%C A127670 Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
%C A127670 The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
%C A127670 Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
%C A127670 The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - _Mike Zabrocki_, Dec 31 2019
%D A127670 Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf.
%D A127670 G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
%D A127670 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.
%H A127670 Vincenzo Librandi, <a href="/A127670/b127670.txt">Table of n, a(n) for n = 1..200</a>
%H A127670 Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barequet/barequet2.html">Proper n-Cell Polycubes in n - 3 Dimensions</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
%H A127670 Mohammad K. Azarian, <a href="http://www.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, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
%H A127670 R. Barequet, G. Barequet, and G. Rote, <a href="http://page.mi.fu-berlin.de/rote/Papers/pdf/Formulae+and+growth+rates+of+high-dimensional+polycubes.pdf">Formulae and growth rates of high-dimensional polycubes</a>, Combinatorica 30 (2010), pp. 257-275.
%H A127670 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez2/florez8.html">Star of David and other patterns in the Hosoya-like polynomials triangles</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
%H A127670 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), Paper No. A14.
%H A127670 Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264">The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.
%H A127670 Rigoberto Flórez, N. McAnally, and A. Mukherjees, <a href="http://math.colgate.edu/~integers/s18b2/s18b2.Abstract.html">Identities for the generalized Fibonacci polynomial</a>, Integers, 18B (2018), Paper No. A2.
%H A127670 M. Haiman, <a href="https://math.berkeley.edu/~mhaiman/ftp/diagonal/diagonal.pdf">Conjectures on the quotient ring by diagonal invariants</a>, preprint, 1993.
%H A127670 M. Haiman, <a href="https://doi.org/10.1023/A:1022450120589">Conjectures on the quotient ring by diagonal invariants</a>, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
%H A127670 O. Khorunzhiy, <a href="https://arxiv.org/abs/2207.00766">Enumeration of tree-type diagrams assembled from oriented chains of edges</a>, arXiv:2207.00766 [math.CO], 2022.
%H A127670 Andrew Snowden, <a href="https://arxiv.org/abs/2302.08699">Measures for the colored circle</a>, arXiv:2302.08699 [math.CO], 2023.
%H A127670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Discriminant.html">Discriminant</a>
%H A127670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%H A127670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%F A127670 a(n) = ((n+1)^(n-2))*2^n, n >= 1.
%F A127670 a(n) = (Det(Vn(xn[1],...,xn[n])))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
%F A127670 a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
%F A127670 a(n) = A007830(n-2)*A000079(n), n >= 2. - _Omar E. Pol_, Aug 27 2011
%F A127670 E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - _Vaclav Kotesovec_, Jun 22 2014
%e A127670 n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 2011
%t A127670 Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* _Vincenzo Librandi_, Jun 23 2014 *)
%o A127670 (Magma) [((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // _Vincenzo Librandi_, Jun 23 2014
%Y A127670 Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
%Y A127670 Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
%Y A127670 A317403 is essentially the same sequence.
%Y A127670 Diagonal 1 of A195739.
%K A127670 nonn,easy
%O A127670 1,2
%A A127670 _Wolfdieter Lang_, Jan 23 2007
%E A127670 Slightly edited by _Gill Barequet_, May 24 2011