%I A123742
%S A123742 1,1,2,48,30240,1596672000,18172937502720000,
%T A123742 122457316443772566896640000,1284319496829094129116119090331648000000,
%U A123742 55603466527142141932748234118927499493985767915520000000
%V A123742 1,-1,-2,48,30240,-1596672000,-18172937502720000,122457316443772566896640000,
%W A123742 1284319496829094129116119090331648000000,
%X A123742 -55603466527142141932748234118927499493985767915520000000
%N A123742 Certain Vandermonde determinants with Fibonacci numbers.
%C A123742 The determinant of a Vandermonde matrix V_n with elements V_n[i,j]=(x_i)^(n-j), 
               i,j,=1..n, is VdmI([x_1,...,x_n]) := Det(V_n)=product(x_i - x_j, 
               1<=i<j<=n) if n>=2. For n=1, Det(V_1)=1. The number of factors for 
               n>=2 is n*(n-1)/2 = A000217(n-1) (triangular numbers).
%C A123742 The signs are +1 for n=1 and (-1)^t(n) with the triangular numbers t(n):=A000217(n-1) 
               for n>=2. Periodic pattern --++, from n=2 on.
%C A123742 The recurrence below follows from the fact that ((-1)^(n-1))*A123741(n-1), 
               n>=2, is the product of the factors of Det(V_n)/Det(V_(n-1)) in the 
               Fibonacci case.
%F A123742 a(n)=VdmI([F(2),F(3),...,F(n+1)]) := Det(V_n[i,j]) with the Vandermonde 
               matrixelements V_n[i,j]:=F(i+1)^(n-j), i,j,=1..n and F(k):=A000045(k) 
               (Fibonacci).
%F A123742 Recurrence: a(n)= ((-1)^(n-1))* A123740(n-1)*a(n-1), a(2):=-1. a(1):=+1.
%e A123742 n=4: V_4=matrix([1,1,1,1],[8,4,2,1],[27,9,3,1],[125,25,5,1]), a(4)=Det(V_4)=+48.
%e A123742 n=4: +48 = a(4) = A123740(3)*a(3) = 24*2.
%Y A123742 Sequence in context: A056989 A090770 A081960 this_sequence A098694 A137592 
               A057527
%Y A123742 Adjacent sequences: A123739 A123740 A123741 this_sequence A123743 A123744 
               A123745
%K A123742 sign,easy
%O A123742 1,3
%A A123742 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 13 2006