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A003112 Permanent of Schur's matrix of order 2n+1.
(Formerly M2509)
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)



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.


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


a(n) = (-1)^n * (2*n+1) * (A003109(n) - A003110(n)). - Sean A. Irvine, Jan 31 2015


(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


Cf. A003109, A003110.

Sequence in context: A279310 A103081 A234600 * A130187 A289488 A054266

Adjacent sequences:  A003109 A003110 A003111 * A003113 A003114 A003115




N. J. A. Sloane


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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Last modified March 31 21:49 EDT 2020. Contains 333151 sequences. (Running on oeis4.)