The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A123742 Certain Vandermonde determinants with Fibonacci numbers. 3
 1, -1, -2, 48, 30240, -1596672000, -18172937502720000, 122457316443772566896640000, 1284319496829094129116119090331648000000, -55603466527142141932748234118927499493985767915520000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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=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. See A203311 for the unsigned version.  [From Clark Kimberling, Jan 03 2012] LINKS FORMULA 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. EXAMPLE 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. MATHEMATICA (See A203311.) CROSSREFS Cf. A203311. Sequence in context: A230886 A090770 A081960 * A203311 A343694 A295177 Adjacent sequences:  A123739 A123740 A123741 * A123743 A123744 A123745 KEYWORD sign,easy AUTHOR Wolfdieter Lang, Oct 13 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 11:03 EST 2021. Contains 349580 sequences. (Running on oeis4.)