A127670 Discriminants of Chebyshev S-polynomials A049310.

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Author

Wolfdieter Lang, Jan 23 2007

Keywords

• nonn
• easy

Comments

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).

From Rigoberto Florez, Sep 02 2018: (Start)

a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).

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.

The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).

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)

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

Examples

			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
		

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 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.
• 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, and Gunter Rote, Proper n-Cell 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. 251-257. 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 high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
• Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like 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.
• Rigoberto 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, 17--76.
• O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
• Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, Discriminant
• Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
• Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Crossrefs

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 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.

Diagonal 1 of A195739.

Programs

Magma

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

Mathematica

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

Formula

a(n) = ((n+1)^(n-2))*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..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).

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.

a(n) = A007830(n-2)*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

Extensions

Slightly edited by Gill Barequet, May 24 2011