A318173 The determinant 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).

2, -11, 158, -6513, 202790, -12710761, 578257422, -45608219247, 8774909485920, -579515898830751, 115918088707226940, -16737522590543449641, 1282860173728469083872, -189053227741259934603831, 55171097827950314187327460, -16235234399834578732807710581

Offset: 1

Author

Stefano Spezia, Aug 20 2018

Keywords

• sign

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 permanent of the matrix M(n) is A306457(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 determinant Det(M(1)) = 2.
For n = 2 the matrix M(2) is
   2, 3
   5, 2
with Det(M(2)) = -11.
For n = 3 the matrix M(3) is
   2, 3, 5
   7, 2, 3
  11, 7, 2
with Det(M(3)) = 158.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..302
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A005843, A000040, A007504, A306457, A306192.

Programs

Maple

f:= proc(n) uses LinearAlgebra;
Determinant(ToeplitzMatrix([seq(ithprime(i),i=2*n-1..n+1,-1),seq(ithprime(i),i=1..n)]))
end proc:
map(f, [$1..20]); # Robert Israel, Aug 30 2018

Mathematica

p[i_]:=Prime[i]; a[n_]:=Det[ToeplitzMatrix[Join[{p[1]},Array[p,n-1,{n+1,2*n-1}]],Array[p,n]]]; Array[a,20]

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) = matdet(tm(n)); \\ Michel Marcus, Mar 17 2019