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

Author

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

Keywords

• nonn
• easy

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

• 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.

Crossrefs

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

Programs

Magma

[0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Mathematica

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

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))

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.

Extensions

Formula and more terms from Vladeta Jovovic, Aug 07 2003