A347768 Permanent of the n X n matrix with (j,k)-entry |j-k+1| (j,k = 1..n).

1, 1, 8, 80, 1568, 42308, 1632544, 82377984, 5340729728, 429905599744, 42164608801024, 4944386388782080, 683353973472423936, 109907353260811403520, 20352830852731108528128, 4299139435513999926820864, 1027450150728092835655335936, 275824741022588671077713641472

Offset: 1

Author

Zhi-Wei Sun, Sep 12 2021

Keywords

• nonn

Comments

Conjecture: For any odd prime p, we have a(p) == 1/2 (mod p).

It is easy to show that for any integer n > 1 the determinant of the n X n matrix with (j,k)-entry |j-k+1| (j,k=1..n) has the value 2^(n-2).

Examples

			a(2) = 1 since the permanent of the matrix [|1-1+1|,|1-2+1|; |2-1+1|,|2-2+1|] = [1,0;2,1] has the value 1.
		

Links

• Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Crossrefs

Cf. A085807.

Programs

Mathematica

a[n_]:=a[n]=Permanent[Table[Abs[j-k+1],{j,1,n},{k,1,n}]]
Table[a[n],{n,1,22}]

PARI

a(n) = matpermanent(matrix(n, n, j, k, abs(j-k+1))); \\ Michel Marcus, Sep 13 2021

Python

from sympy import Matrix
def A347768(n): return Matrix(n,n,[abs(j-k+1) for j in range(n) for k in range(n)]).per() # Chai Wah Wu, Sep 14 2021