A007701 a(0) = 0; for n > 0, a(n) = n^n*2^((n-1)^2). (Formerly M4585)

0, 1, 8, 432, 131072, 204800000, 1565515579392, 56593444029595648, 9444732965739290427392, 7146646609494406531041460224, 24178516392292583494123520000000000

Offset

0,3

Comments

Discriminant of Chebyshev polynomial T_n (x) of first kind.

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Delbert L. Johnson, Table of n, a(n) for n = 0..55

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Chebyshev polynomials.

Formula

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

a(n) = ((2^((n-1)^2))*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((2*i-1)*Pi/(2*n)), i=1..n, are the zeros of the Chebyshev T(n,x) polynomials.

a(n) = ((-1)^(n*(n-1)/2))*(2^((n-1)*(n-2))) * Product_{i=1..n} ((d/dx)T(n,x)|_{x=xn[i]}), n > 0, with the zeros xn[i], i=1..n, given above.

Mathematica

Join[{0}, Table[n^n 2^(n-1)^2, {n, 10}]] (* Harvey P. Dale, Sep 04 2023 *)

Prog

(PARI) a(n)=if(n<1, 0, n^n*2^((n-1)^2))
(PARI) a(n)=if(n<1, 0, poldisc(poltchebi(n)))

Crossrefs

Cf. A086804.

Cf. A127670 (discriminant for S-polynomials).

Sequence in context: A132099 A186419 A371560 * A101356 A231102 A231824

Adjacent sequences: A007698 A007699 A007700 * A007702 A007703 A007704

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Somos, Jun 26 2002

Status

approved