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A318173
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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).
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13
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2, -11, 158, -6513, 202790, -12710761, 578257422, -45608219247, 8774909485920, -579515898830751, 115918088707226940, -16737522590543449641, 1282860173728469083872, -189053227741259934603831, 55171097827950314187327460, -16235234399834578732807710581
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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]
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PROG
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(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; }
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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