A085807 Permanent of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

1, 0, 1, 4, 64, 1152, 34372, 1335008, 69599744, 4577345152, 374491314176, 37154032517376, 4402467119882240, 613680867638476800, 99443966100565999872, 18534733913629064343552, 3937496200758879526977536, 945776134421421651222708224, 255043190756805184245158084608

Offset: 0

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 24 2003

Keywords

• nonn

Comments

Conjecture: For any odd prime p, we have a(p) == -1/2 (mod p). - Zhi-Wei Sun, Aug 30 2021

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal. - Stefano Spezia, Jul 05 2024

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..37
• Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Crossrefs

Cf. A085750, A204249, A278845, A278847.

Programs

Maple

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> abs(i-j)))):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 14 2016

Mathematica

a[n_]:=Permanent[Table[Abs[i - j], {i, n}, {j, n}]]; Join[{1}, Array[a, 18]] (* Stefano Spezia, Jun 28 2024 *)

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(n=1,22,a=matrix(n,n,i,j,abs(i-j));print1(permRWNb(a)",")) \\  Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

PARI

{a(n) = matpermanent(matrix(n, n, i, j, abs(i-j)))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021

Python

from sympy import Matrix
def A085807(n): return Matrix(n,n,[abs(j-k) for j in range(n) for k in range(n)]).per() # Chai Wah Wu, Sep 14 2021

Extensions

More terms from Vladeta Jovovic, Jul 26 2003

a(0)=1 prepended by Alois P. Heinz, Nov 14 2016