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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).

%I A318173 #41 Mar 27 2019 03:59:20
%S A318173 2,-11,158,-6513,202790,-12710761,578257422,-45608219247,
%T A318173 8774909485920,-579515898830751,115918088707226940,
%U A318173 -16737522590543449641,1282860173728469083872,-189053227741259934603831,55171097827950314187327460,-16235234399834578732807710581
%N 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).
%C A318173 The trace of the matrix M(n) is A005843(n).
%C A318173 The sum of the first row of the matrix M(n) is A007504(n).
%C A318173 The permanent of the matrix M(n) is A306457(n).
%C A318173 For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).
%H A318173 Robert Israel, <a href="/A318173/b318173.txt">Table of n, a(n) for n = 1..302</a>
%H A318173 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A318173 For n = 1 the matrix M(1) is
%e A318173    2
%e A318173 with determinant Det(M(1)) = 2.
%e A318173 For n = 2 the matrix M(2) is
%e A318173    2, 3
%e A318173    5, 2
%e A318173 with Det(M(2)) = -11.
%e A318173 For n = 3 the matrix M(3) is
%e A318173    2, 3, 5
%e A318173    7, 2, 3
%e A318173   11, 7, 2
%e A318173 with Det(M(3)) = 158.
%p A318173 f:= proc(n) uses LinearAlgebra;
%p A318173 Determinant(ToeplitzMatrix([seq(ithprime(i),i=2*n-1..n+1,-1),seq(ithprime(i),i=1..n)]))
%p A318173 end proc:
%p A318173 map(f, [$1..20]); # _Robert Israel_, Aug 30 2018
%t A318173 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]
%o A318173 (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;}
%o A318173 a(n) = matdet(tm(n)); \\ _Michel Marcus_, Mar 17 2019
%Y A318173 Cf. A005843, A000040, A007504, A306457, A306192.
%K A318173 sign
%O A318173 1,1
%A A318173 _Stefano Spezia_, Aug 20 2018