  Exceptional orthogonal Laguerre polynomials can be viewed as an extension of
the classical Laguerre polynomials per excluding polynomials of certain
order(s) from being eigenfunctions for the corresponding exceptional
differential operator. We are interested in the (so-called) Type I
$X_1$-Laguerre polynomial sequence $\{L_n^\alpha\}_{n=1}^\infty$, $\text{deg}
\,p_n = n$ and $\alpha>0$, where the constant polynomial is omitted.
  We derive two representations for the polynomials in terms of moments by
using determinants. The first representation in terms of the canonical moments
is rather cumbersome. We introduce adjusted moments and find a second, more
elegant formula. We deduce a recursion formula for the moments and the adjusted
ones. The adjusted moments are also expressed via a generating function. We
observe a certain detachedness of the first two moments from the others.
