A123743 - OEIS

%I #5 Oct 02 2025 21:11:46

%S 1,2,12,1440,7257600,4981616640000,1190690865178214400000,

%T 272795714695463271824306995200000,

%U 157357907118293002525216789633250308915200000000

%N Certain Vandermonde determinants with Fibonacci numbers.

%C The determinant of a Vandermonde matrix VM_n with elements VM_n[i,j]=(x_j)^i, i,j,=1..n, is VdmII([x_1,...,x_n]) := Det(VM_n)= product(x_k,k=1,...,n)*product(x_j - x_i, 1<=i<j<=n) if n>=2. For n=1, Det(VM_1)=1.

%F a(n)= Fibfac(n)* |A123742(n)|, with the Fibonacci factorials Fibfac(n):=A003266(n+1).

%F a(n)=VdmII([F(2),F(3),...,F(n+1)]) := Det(VM_n[i,j]) with the Vandermonde matrix elements VM_n[i,j]:=F(j+1)^i, i,j,=1..n and F(k):=A000045(k) (Fibonacci).

%e n=4: VM_4 = matrix([1,2,3,5],[1,4,9,25],[1,8,27,125],[1,16,81,625]).

%e a(4)=Det(VM_4) = 1440 = 30*48 = A003266(5)*|A123742(4)|.

%Y Cf. A123742 (another version).

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Oct 13 2006