  The enumeration of standard Young tableaux (SYTs) of shape {\lambda} can be
easily computed by the hook-length formula. In 1981, Amitai Regev proved that
the number of SYTs having at most three rows with n entries equals the nth
Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of
skew shape {\lambda}/(2, 1) over all partitions {\lambda} having at most three
parts with n entries is the difference of two Motzkin numbers, M_{n-1} -
M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computer
program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of
skew shape with at most three rows to prove Regev's conjecture, and Eu also
indirectly showed that for the fixed {\mu} = ({\mu}1,{\mu}2) the number of SYTs
of skew shape {\lambda}/{\mu} over all partitions {\lambda} having at most
three parts can be expressed as a linear combination of the Motzkin numbers. In
this paper, we will find an explicit formula for the generating function for
the general case: for each partition {\mu} having at most three parts the
generating function gives a formula for the coefficients of the linear
combination of Motzkin numbers. We will also show that these generating
functions are unexpectedly related to the Chebyshev polynomials of the second
kind.
