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].
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, Jun 26 2002
STATUS
approved