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%I #33 Feb 04 2024 18:37:20
%S 1,2,1,-2,0,288,-1728,-26240,222272,1636864,-8434688,-61820416,
%T 238704640,544024576,3294658560,-71814283264,359994671104,
%U 17294535000064,302441193013248,-2311203985948672,-11313883306262528,-31078379553816576,26574426771056230400
%N Determinant of Hankel matrix of the first 2n-1 prime numbers.
%C Determinant of n X n matrix with entries prime(X+Y-1).
%C a(0) = 1 by convention.
%C I conjecture that a(4) is the only zero. - _Jon Perry_, Mar 22 2004
%H Klaus Brockhaus, <a href="/A024356/b024356.txt">Table of n, a(n) for n = 0..200</a>
%e a(2) = 1 because det[[2,3],[3,5]] = 1.
%e From _Klaus Brockhaus_, May 12 2010: (Start)
%e a(5) = determinant(M) = 288 where M is the matrix
%e [ 2 3 5 7 11]
%e [ 3 5 7 11 13]
%e [ 5 7 11 13 17]
%e [ 7 11 13 17 19]
%e [11 13 17 19 23] . (End)
%t a[n_]:=Det[Table[Prime[i+j-1],{i,n},{j,n}]]; Join[{1},Array[a, 20]] (* _Stefano Spezia_, Feb 03 2024 *)
%o (PARI) for (i=0,20,print1(","matdet(matrix(i,i,X,Y,prime(X+Y-1))))) \\ _Jon Perry_, Mar 22 2004
%o (Magma) Hankel_prime:=function(n); M:=ScalarMatrix(n, 0); for j in [1..n] do for k in [1..n] do M[j, k]:=NthPrime(j+k-1); end for; end for; return M; end function; [ Determinant(Hankel_prime(n)): n in [0..22] ];
%o [1] cat [ Determinant( SymmetricMatrix( &cat[ [ NthPrime(j+k-1): k in [1..j] ]: j in [1..n] ] ) ): n in [1..22] ]; // _Klaus Brockhaus_, May 12 2010
%Y Cf. A290302.
%K sign
%O 0,2
%A _Jeffrey Shallit_, Jun 08 2000
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