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a(0) = 0; for n > 0, a(n) = n^n*2^((n-1)^2).
(Formerly M4585)
9

%I M4585 #26 Sep 04 2023 12:53:29

%S 0,1,8,432,131072,204800000,1565515579392,56593444029595648,

%T 9444732965739290427392,7146646609494406531041460224,

%U 24178516392292583494123520000000000

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

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

%D 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.

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

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

%H Delbert L. Johnson, <a href="/A007701/b007701.txt">Table of n, a(n) for n = 0..55</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

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

%F 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.

%F 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.

%t Join[{0},Table[n^n 2^(n-1)^2,{n,10}]] (* _Harvey P. Dale_, Sep 04 2023 *)

%o (PARI) a(n)=if(n<1,0,n^n*2^((n-1)^2))

%o (PARI) a(n)=if(n<1,0,poldisc(poltchebi(n)))

%Y Cf. A086804.

%Y Cf. A127670 (discriminant for S-polynomials).

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Additional comments from _Michael Somos_, Jun 26 2002