|
%I M2509 #60 Aug 19 2021 01:48:49
%S 1,-3,-5,-105,81,6765,175747,30375,25219857,142901109,4548104883,
%T -31152650265,-5198937484375,65230244418933,-1300425712598285,
%U 126691467546591,868088125376401545,-15139017417029296875
%N Permanent of Schur's matrix of order 2n+1.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 121.
%H R. L. Graham and D. H. Lehmer, <a href="http://www.math.ucsd.edu/~ronspubs/76_07_schurs_matrix.pdf">On the Permanent of Schur's Matrix</a>, Jour. Australian Math. Soc. 21 (series A) (1976), 487-497.
%H R. L. Graham and D. H. Lehmer, <a href="/A003112/a003112.png">On the Permanent of Schur's Matrix</a>, 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]
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1016/0022-314X(73)90056-5">Some properties of circulants</a>, J. Number Theory 5 (1973), 43-54. (See page 48.)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchurMatrix.html">Schur Matrix</a>
%F a(n) = (-1)^n * (2*n+1) * (A003109(n) - A003110(n)). - _Sean A. Irvine_, Jan 31 2015
%t GrayInsert[n_] := Block[{q = n, j = 1}, While[ EvenQ[q], q /= 2; j++]; {j, (-1)^((q - 1)/2)}];abs2[x_] := Re[x]^2 + Im[x]^2;Schur[n_, prec_] := Block[{xi = N[E^(2 Pi* I/n), prec], m, i, j, rowsum, sum = 0}, m = Table[xi^Mod[i j, n], {i, n - 2}, {j, (n - 1)/2}]; rowsum = Table[xi^(-j) + N[1/2, prec], {j, (n - 1)/2}]; sum = abs2[Times @@ rowsum]; Do[gi = GrayInsert[i]; rowsum += gi[[2]]* m[[gi[[1]]]]; sum += N[(-1)^i* abs2[Times @@ rowsum], prec], {i, 2^(n - 2) - 1}]; -Round[n *2* sum]] /; OddQ[n]; Do[ Print[{n, Schur[n, n+1]}], {n, 1, 16}] (* copied the necessary Mathematica coding from Prof. Ilan Vardi, _Robert G. Wilson v_, Apr 19 2020 *)
%o (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)
%o for(k=1,12,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
%o (PARI) for(k=1, 12, a=matrix(2*k-1, 2*k-1, i, j, exp(2*Pi*I*(i-1)*(j-1)/(2*k-1))); print1(round(real(matpermanent(a)))", ")) \\ _Vaclav Kotesovec_, Aug 12 2021
%Y Cf. A003109, A003110.
%K hard,more,sign
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 17 2007
%E a(15)-a(16) from _Vaclav Kotesovec_, Dec 11 2013
%E a(17) from _Vaclav Kotesovec_, Aug 19 2021
|
