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A003112 Permanent of Schur's matrix of order 2n+1.
(Formerly M2509)
4
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)
OFFSET

0,2

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

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

FORMULA

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

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 *)

PROG

(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

CROSSREFS

Cf. A003109, A003110.

Sequence in context: A279310 A103081 A234600 * A130187 A289488 A054266

Adjacent sequences:  A003109 A003110 A003111 * A003113 A003114 A003115

KEYWORD

hard,more,sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

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

STATUS

approved

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Last modified May 29 15:44 EDT 2020. Contains 334704 sequences. (Running on oeis4.)