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A080358 Value of Vandermonde determinant for the first n prime numbers: V[prime(1), ..., prime(n)].

%I A080358 #31 Apr 13 2024 11:46:01
%S A080358 1,1,1,6,240,414720,4379443200,2648687247360000,
%T A080358 11619303595714805760000,4047756373260469165621248000000,
%U A080358 311107430628520522709128328175943680000000,152539657943794787580793302587123569672794931200000000
%N A080358 Value of Vandermonde determinant for the first n prime numbers: V[prime(1), ..., prime(n)].
%C A080358 The value of the Vandermonde determinant is unchanged if the numbers are shifted by an arbitrary constant c, i.e., V[prime(1), ..., prime(n)] = V[prime(1)-c, ..., prime(n)-c].
%C A080358 For a guide to related sequences, see A093883. - _Clark Kimberling_, Jan 03 2012
%H A080358 Alois P. Heinz, <a href="/A080358/b080358.txt">Table of n, a(n) for n = 0..36</a>
%F A080358 a(n) = Product_{i, j, i>j} (prime(i) - prime(j)). a(n) is the product of binomial(n, 2) prime differences of not necessarily consecutive primes.
%F A080358 For n > 1, a(n) = sqrt(Delta_n), where Delta_n is the discriminant of the polynomial (x - 2)*(x - 3)*...*(x - prime(n)). - _Thomas Ordowski_, Mar 15 2023
%e A080358 a(1)=1 corresponds to 1 X 1 V-matrix, while a(2)=1 is computed from a 2 X 2 matrix.
%e A080358 n = 2: a(2) = prime(2) - prime(1) = 3 - 2 = 1;
%e A080358 n = 3: a(3) = (5-3)*(5-2)*(3-2) = 2*3*1 = 6; ...
%e A080358 n = 6: a(6) = (13-11)*(13-7)*(13-5)*(13-3)*(13-2)*(11-7)*(11-5)*(11-3)*(11-2)*(7-5)*(7-3)*(7-2)*(5-3)*(5-2)*(3-2) = 2*6*8*10*11*4*6*8*9*2*4*5*2*3*1 = 10560*1728*40*6*1 = 4379443200.
%p A080358 with(LinearAlgebra):
%p A080358 a:= n-> Determinant(Matrix(n, (i,j)-> ithprime(i)^(j-1))):
%p A080358 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jul 22 2017
%t A080358 b[x_] := Prime[x] d[x_] := b[x+1]-b[x] t[m_] := b[m+1]-Table[b[x], {x, 1, m}] pt[x_] := Apply[Times, t[x]] va[x_] := Apply[Times, Table[pt[w], {w, 1, x}]] Table[va[j], {j, 1, 10}]
%Y A080358 Cf. A000040, A001359, A203521, A290179.
%K A080358 nonn
%O A080358 0,4
%A A080358 _Labos Elemer_, Feb 19 2003