A306457 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

1, 2, 19, 546, 40851, 4747510, 986799301, 292666754602, 135134321711681, 80312872924339660, 55242523096584443271, 52058868505260739019880, 55579419798019716586180451, 72402676504369062268839297084, 120521257466525185305708420453019, 234000358527930078723939842673115488

Offset: 0

Author

Stefano Spezia, Feb 17 2019

Keywords

• nonn

Comments

The trace of the matrix M(n) is A005843(n).

The sum of the first row of the matrix M(n) is A007504(n).

The determinant of the matrix M(n) is A318173(n).

For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).

Examples

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    5, 2
with permanent a(2) = 19.
For n = 3 the matrix M(3) is
    2, 3, 5
    7, 2, 3
   11, 7, 2
with permanent a(3) = 546.
		

Links

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

Crossrefs

Cf. A005843, A000040, A007504, A318173, A306192.

Programs

Maple

f:= proc(n) uses LinearAlgebra; `if`(n=0, 1, Permanent(ToeplitzMatrix([seq(ithprime(i), i=2*n-1..n+1, -1), seq(ithprime(i), i=1..n)]))) end proc: map(f, [$0..15]);

Mathematica

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

PARI

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(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)); \\ Michel Marcus, Mar 16 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 06 2019