A318173 - OEIS

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

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OFFSET

1,1

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).

LINKS

Robert Israel, Table of n, a(n) for n = 1..302

Wikipedia, Toeplitz Matrix

EXAMPLE

For n = 1 the matrix M(1) is

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.

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]

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

CROSSREFS

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

Sequence in context: A058154 A275923 A288560 * A349639 A067968 A295269

Adjacent sequences: A318170 A318171 A318172 * A318174 A318175 A318176

KEYWORD

sign

AUTHOR

Stefano Spezia, Aug 20 2018

STATUS

approved