

A127670


Discriminants of Chebyshev Spolynomials A049310.


21



1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104
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OFFSET

1,2


COMMENTS

a(n1) is the number of fixed ncell polycubes that are proper in n1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n1) is the discriminant of the MorganVoyce Fibonaccitype polynomial B(n).
MorganVoyce Fibonaccitype polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n1)  B(n2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n1) + F(n2) for n > 1. (End)
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


REFERENCES

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.
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. 1922, 2015.
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.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, Gunter Rote, Proper nCell Polycubes in n  3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From N. J. A. Sloane, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of highdimensional polycubes, Combinatorica 30 (2010), pp. 257275.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoyalike polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 1776.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, MorganVoyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial


FORMULA

a(n) = ((n+1)^(n2))*2^n, n >= 1.
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..n1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((1)^(n*(n1)/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.
a(n) = A007830(n2)*A000079(n), n >= 2.  Omar E. Pol, Aug 27 2011
E.g.f.: LambertW(2*x)*(2+LambertW(2*x))/(4*x).  Vaclav Kotesovec, Jun 22 2014


EXAMPLE

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


MATHEMATICA

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)


PROG

(MAGMA) [((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014


CROSSREFS

Cf. A007701 (Tpolynomials), A086804 (Upolynomials), A171860 and A191092 (fixed ncell polycubes proper in n2 and n3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Sequence in context: A325574 A113131 A195762 * A317403 A243468 A317677
Adjacent sequences: A127667 A127668 A127669 * A127671 A127672 A127673


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Jan 23 2007


EXTENSIONS

Slightly edited by Gill Barequet, May 24 2011


STATUS

approved



