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A127670
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Discriminants of Chebyshev S-polynomials A049310.
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7
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1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
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REFERENCES
| R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275. See Th. 1. From N. J. A. Sloane, Oct 16 2010.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.
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FORMULA
| a(n)= ((n+1)^{n-2})*2^n, n>=1.
a(n)= (Det(Vn(xn[1],..,xn[n]))^2 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]:=2*cos(Pi*i/(n+1)), i=1,..,n, are the zeros of S(n,x):=U(n,x/2).
a(n)= ((-1)^(n*(n-1)/2))*product(diff(S(n,x))|_{x=xn[j]},j=1..n)), n>=1, with the zeros xn[j],j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
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EXAMPLE
| n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],..,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). [From Wolfdieter Lang, Aug 07 2011]
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CROSSREFS
| Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.)
Sequence in context: A005263 A113131 A195762 * A191459 A184359 A005172
Adjacent sequences: A127667 A127668 A127669 * A127671 A127672 A127673
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 23 2007
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EXTENSIONS
| Slightly edited by Gill Barequet (barequet(AT)cs.technion.ac.il), May 24 2011
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