1,3

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).

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.

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.

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).

Recurrence: a(n)= ((-1)^(n-1))* A123740(n-1)*a(n-1), a(2):=-1. a(1):=+1.

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.

n=4: +48 = a(4) = A123740(3)*a(3) = 24*2.

Sequence in context: A056989 A090770 A081960 this_sequence A098694 A137592 A057527

Adjacent sequences: A123739 A123740 A123741 this_sequence A123743 A123744 A123745

sign,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 13 2006