A307783 The permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of n, n-1, ..., 1.

1, 5, 62, 1472, 57228, 3300052, 264163120, 28004426240, 3796084024832, 640290996560896, 131495036625989504, 32300689159458652160, 9350873610168606862080, 3150550820854335942423808, 1222211647879605626853439488, 540858935979668390014623285248, 270804098518125729769134021574656

Offset: 1

Author

Stefano Spezia, Apr 28 2019

Keywords

• nonn

Comments

The matrix M(n) differs from that of A204235 in using for the first row the positive integers 1, 2,..., n in decreasing order in place of in increasing order (see examples).

The trace of the matrix M(n) is A000290(n).

The determinant of the matrix M(n) is A001792(n-1).

The sum of the k-th row of the matrix M(n) is A008867(n,k).

For n > k, the sum of the k-diagonal of the matrix M(n) is A055461(n,k).

Examples

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  2, 1
  1, 2
with permanent a(2) = 5.
For n = 3 the matrix M(3) is
  3, 2, 1
  2, 3, 2
  1, 2, 3
with permanent a(3) = 62.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..35
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A000290, A001792, A008867, A055461, A204235.

Programs

Maple

f:= proc(n) uses LinearAlgebra; Permanent(ToeplitzMatrix([i, i=n..1, -1)])) end proc: map(f, [$1..17]);

Mathematica

b[i_]:=i; a[n_]:=Permanent[ToeplitzMatrix[Reverse[Array[b, n]], Reverse[Array[b, n ]]]]; Array[a, 17]

PARI

{a(n) = matpermanent(matrix(n, n, i, j, n + 1 - max(i - j + 1, j - i + 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Apr 29 2019