A123743 Certain Vandermonde determinants with Fibonacci numbers.

1, 2, 12, 1440, 7257600, 4981616640000, 1190690865178214400000, 272795714695463271824306995200000, 157357907118293002525216789633250308915200000000

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..9.

Formula

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

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).

Example

n=4: VM_4 = matrix([1,2,3,5],[1,4,9,25],[1,8,27,125],[1,16,81,625]).
a(4)=Det(VM_4) = 1440 = 30*48 = A003266(5)*|A123742(4)|.

Crossrefs

Cf. A123742 (another version).

Sequence in context: A081701 A069714 A176391 * A111180 A235857 A085912

Adjacent sequences: A123740 A123741 A123742 * A123744 A123745 A123746

Keyword

nonn,easy

Author

Wolfdieter Lang Oct 13 2006

Status

approved