A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

0, 1, 16, 2048, 1638400, 7247757312, 164995463643136, 18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, 271732164163901599116133024293512544256

Offset

0,3

Comments

Discriminant of Chebyshev polynomial U_n (x) of second kind.

Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.

The absolute value of the discriminant of Pell polynomials is a(n-1).

Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - Rigoberto Florez, Sep 01 2018

References

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

Links

Table of n, a(n) for n=0..10.

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, 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.

R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.

R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Eric Weisstein's World of Mathematics, Discriminant

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind

Eric Weisstein's World of Mathematics, Pell Polynomial

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.

a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, 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]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.

a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

Mathematica

Join[{0}, Table[(n+1)^(n-2) 2^n^2, {n, 10}]] (* Harvey P. Dale, May 01 2015 *)

Prog

(PARI) a(n)=if(n<1, 0, (n+1)^(n-2)*2^(n^2))
(PARI) a(n)=if(n<1, 0, n++; poldisc(poltchebi(n)'/n))
(Magma) [0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Crossrefs

Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Sequence in context: A176886 A159389 A291828 * A317450 A121366 A101800

Adjacent sequences: A086801 A086802 A086803 * A086805 A086806 A086807

Keyword

nonn,easy

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Extensions

Formula and more terms from Vladeta Jovovic, Aug 07 2003

Status

approved