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A123742 Certain Vandermonde determinants with Fibonacci numbers. 3


%S 1,-1,-2,48,30240,-1596672000,-18172937502720000,

%T 122457316443772566896640000,1284319496829094129116119090331648000000,

%U -55603466527142141932748234118927499493985767915520000000

%N Certain Vandermonde determinants with Fibonacci numbers.

%C 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 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 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.

%C See A203311 for the unsigned version. [From Clark Kimberling, Jan 03 2012]

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

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

%t (See A203311.)

%Y Cf. A203311.

%K sign,easy

%O 1,3

%A _Wolfdieter Lang_, Oct 13 2006

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