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A024356 Determinant of Hankel matrix of the first 2n-1 prime numbers.

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