On some new sums of Fibonomial coefficients (Q2894234)

Let \(F_n\) be the \(n\)th Fibonacci number. The Fibonomial coefficients \({n\brack k}_F\) are defined for \(n\geq k > 0\) as follows NEWLINE\[NEWLINE{n\brack k}_F = \frac{F_nF_{n-1}\cdots F_{n -k+1}}{F_1F_2\cdots F_k}NEWLINE\]NEWLINE with \({n\brack 0}_F= 1\) and \({n\brack 0}_F= 0\) for \(n < k\). In this paper, the authors find some interesting sums among Fibonomial coefficients. In particular, they prove that NEWLINE\[NEWLINE\sum_{j=0}^{4m+2} (-1)^{\frac j2 (j+1)} {4m+2\brack j}_F F_{n+4m+2-j} = 0,NEWLINE\]NEWLINE is valid for all non-negative integers \(m\) and \(n\).