A007701 a(0) = 0; for n > 0, a(n) = n^n*2^((n-1)^2).
0, 1, 8, 432, 131072, 204800000, 1565515579392, 56593444029595648, 9444732965739290427392, 7146646609494406531041460224, 24178516392292583494123520000000000
Offset: 0
Keywords
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.
Programs
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Mathematica
Join[{0},Table[n^n 2^(n-1)^2,{n,10}]] (* Harvey P. Dale, Sep 04 2023 *) -
PARI
a(n)=if(n<1,0,n^n*2^((n-1)^2))
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PARI
a(n)=if(n<1,0,poldisc(poltchebi(n)))
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.
Extensions
Additional comments from Michael Somos, Jun 26 2002
Comments