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

0,2

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).

Links

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

Wikipedia, Toeplitz Matrix

Example

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.

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]

Prog

(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

Crossrefs

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

Sequence in context: A120420 A350939 A239674 * A158099 A015204 A247241

Adjacent sequences: A306454 A306455 A306456 * A306458 A306459 A306460

Keyword

nonn

Author

Stefano Spezia, Feb 17 2019

Extensions

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

Status

approved