In this paper, we consider a generalized Catalan triangle defined by k(m)/n(2n n - k) for positive integer m. Then we compute the weighted half binomial sums with the certain powers of generalized Fibonacci and Lucas numbers of the form Sigma(n)(k=0) (2n n + k) k(m)/nX(tk)(r), where X-n either generalized Fibonacci or Lucas numbers, t and r are integers for 1 <= m <= 6. After we describe a general methodology to show how to compute the sums for further values of m.