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%I #108 Feb 23 2023 17:08:02
%S 1,4,32,400,6912,153664,4194304,136048896,5120000000,219503494144,
%T 10567230160896,564668382613504,33174037869887488,2125764000000000000,
%U 147573952589676412928,11034809241396899282944,884295678882933431599104
%N Discriminants of Chebyshev S-polynomials A049310.
%C a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
%C From _Rigoberto Florez_, Sep 02 2018: (Start)
%C a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
%C 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 The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
%C 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 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 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 G. Barequet, 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 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 Vincenzo Librandi, <a href="/A127670/b127670.txt">Table of n, a(n) for n = 1..200</a>
%H 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 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 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 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 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 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 R. 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 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 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 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 Andrew Snowden, <a href="https://arxiv.org/abs/2302.08699">Measures for the colored circle</a>, arXiv:2302.08699 [math.CO], 2023.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Discriminant.html">Discriminant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%F a(n) = ((n+1)^(n-2))*2^n, n >= 1.
%F 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 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 a(n) = A007830(n-2)*A000079(n), n >= 2. - _Omar E. Pol_, Aug 27 2011
%F E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - _Vaclav Kotesovec_, Jun 22 2014
%e 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 Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* _Vincenzo Librandi_, Jun 23 2014 *)
%o (Magma) [((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // _Vincenzo Librandi_, Jun 23 2014
%Y Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
%Y Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
%Y A317403 is essentially the same sequence.
%Y Diagonal 1 of A195739.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jan 23 2007
%E Slightly edited by _Gill Barequet_, May 24 2011
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