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A003112
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Permanent of Schur's matrix of order 2n+1.
(Formerly M2509)
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4
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1, -3, -5, -105, 81, 6765, 175747, 30375, 25219857, 142901109, 4548104883, -31152650265, -5198937484375, 65230244418933, -1300425712598285, 126691467546591, 868088125376401545
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 121.
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LINKS
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Table of n, a(n) for n=0..16.
R. L. Graham and D. H. Lehmer, On the Permanent of Schur's Matrix, Jour. Australian Math. Soc. 21 (series A) (1976), 487-497.
R. L. Graham and D. H. Lehmer, On the Permanent of Schur's Matrix, annotated scanned copy of pages 496-497 only. [When Ron Graham showed me the first draft of this article in 1974, I pointed out that he and Dick Lehmer had overlooked the fact that this same sequence had appeared a year earlier in another Lehmer article! - N. J. A. Sloane, Sep 13 2018]
D. H. Lehmer, Some properties of circulants, J. Number Theory 5 (1973), 43-54. (See page 48.)
Eric Weisstein's World of Mathematics, Schur Matrix
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FORMULA
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a(n) = (-1)^n * (2*n+1) * (A003109(n) - A003110(n)). - Sean A. Irvine, Jan 31 2015
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PROG
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(PARI) permRWNb(a)=n=matsize(a)[1]; if(n==1, return(a[1, 1])); sg=1; in=vectorv(n); x=in; x=a[, n]-sum(j=1, n, a[, j])/2; p=prod(i=1, n, x[i]); for(k=1, 2^(n-1)-1, sg=-sg; j=valuation(k, 2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[, j]; p+=prod(i=1, n, x[i], sg)); return(2*(2*(n%2)-1)*p) for(k=1, 14, n=2*k-1; z=exp(2*Pi*I/n); a=matrix(n, n, i, j, z^((i-1)*(j-1))); print1(round(real(permRWNb(a)))", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 17 2007
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CROSSREFS
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Cf. A003109, A003110.
Sequence in context: A279310 A103081 A234600 * A130187 A289488 A054266
Adjacent sequences: A003109 A003110 A003111 * A003113 A003114 A003115
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KEYWORD
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hard,more,sign
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 17 2007
a(15)-a(16) from Vaclav Kotesovec, Dec 11 2013
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STATUS
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approved
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