A formula for the number of up-down words

A word $w_1{w}_2\cdots w_n$ is said to be up-down if $w_1<w_2>w_3\cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we derive a closed-form formula for these numbers.