A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

1, 1, 7, 100, 2840, 129428, 8613997, 791557152, 95921167710, 14818153059968, 2842735387366627, 663020104070865664, 184757202542187563476, 60623405966739216871680, 23135486197103263598936745, 10160292704659539620791062528, 5087671168376607498331875818106

Offset: 0

Author

Stefano Spezia, Dec 30 2018

Keywords

• nonn

Comments

The matrix M(n) differs from that of A306457 in using successive positive integers in place of successive prime numbers. [Modified by Stefano Spezia, Dec 20 2019 at the suggestion of Michel Marcus]

The trace of M(n) is A000027(n).

The sum of the first row of M(n) is A000217(n).

The sum of the first column of M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 19 2019]

For n > 1, the sum of the superdiagonal of M(n) is A005843(n).

For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - Stefano Spezia, Dec 29 2019

For n > 1 and k > 0, the sum of the k-th subdiagonal of M(n) is A120070(n,k). - Stefano Spezia, Dec 31 2019

Examples

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with permanent a(2) = 7.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with permanent a(3) = 100.
		

Links

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

Crossrefs

Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).

Programs

Maple

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
         seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
seq(a(n), n = 0 .. 15);

Mathematica

b[n_]:=n; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]]; Array[a, 15,0]

PARI

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-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 19 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 19 2019