A003112 Permanent of Schur's matrix of order 2n+1.

1, -3, -5, -105, 81, 6765, 175747, 30375, 25219857, 142901109, 4548104883, -31152650265, -5198937484375, 65230244418933, -1300425712598285, 126691467546591, 868088125376401545, -15139017417029296875

Offset: 0

Author

N. J. A. Sloane

Keywords

• hard
• more
• sign

References

• 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.

Links

• 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

Crossrefs

Cf. A003109, A003110.

Programs

Mathematica

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 *)

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=1,12,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

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

Formula

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

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 17 2007

a(15)-a(16) from Vaclav Kotesovec, Dec 11 2013

a(17) from Vaclav Kotesovec, Aug 19 2021