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

0,3

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).

Links

Stefano Spezia, Table of n, a(n) for n = 0..35

Wikipedia, Toeplitz matrix

Example

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.

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]

Prog

(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

Crossrefs

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

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

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

Sequence in context: A347483 A219089 A243683 * A167868 A368190 A238751

Adjacent sequences: A323252 A323253 A323254 * A323256 A323257 A323258

Keyword

nonn

Author

Stefano Spezia, Jan 09 2019

Extensions

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

Status

approved