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A003112
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Permanent of Schur's matrix of order 2n+1.
(Formerly M2509)
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5
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1, -3, -5, -105, 81, 6765, 175747, 30375, 25219857, 142901109, 4548104883, -31152650265, -5198937484375, 65230244418933, -1300425712598285, 126691467546591, 868088125376401545, -15139017417029296875
(list;
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history;
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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..17.
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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MATHEMATICA
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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 *)
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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, 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
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CROSSREFS
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Cf. A003109, A003110.
Sequence in context: A338269 A346710 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
a(17) from Vaclav Kotesovec, Aug 19 2021
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STATUS
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approved
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