A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

1, 1, 11, 248, 9968, 638772, 60061657, 7798036000, 1336715859150, 292406145227392, 79483340339739367, 26280500564448081664, 10386012861097225139356, 4834639222955142417477888, 2618110215141486526589786501, 1631888040186649673361825042432, 1159983453675106278249250918734938

Offset: 0

Author

Stefano Spezia, Jan 09 2019

Keywords

• nonn

Comments

The trace of the matrix M(n) is A000384(n).

The sum of the first row of the matrix M(n) is A034856(n).

The sum of the first column of the matrix M(n) is A000326(n).

For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).

For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

Examples

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with permanent a(3) = 248.
		

Links

• Stefano Spezia, Table of n, a(n) for n = 0..35
• Wikipedia, Toeplitz matrix

Crossrefs

Cf. A000290, A000326, A000384, A001105, A001792.

Cf. A034856, A204235, A318173, A322908, A322909.

Cf. A323254 (determinant of matrix M(n)).

Programs

Mathematica

b[i_]:=i; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n }]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]]; Array[a, 16, 0]

PARI

tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 11 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 08 2019