  Arrowed Gelfand-Tsetlin patterns have recently been introduced to study
alternating sign matrices. In this paper, we show that a $(-1)$-enumeration of
arrowed Gelfand-Tsetlin patterns can be expressed by a simple product formula.
The numbers are a one-parameter generalization of the numbers $2^{n(n-1)/2}
\prod_{j=0}^{n-1} \frac{(4j+2)!}{(n+2j+1)!}$ that appear in recent work of Di
Francesco. A second result concerns the (-1)-enumeration of arrowed
Gelfand-Tsetlin patterns when excluding double-arrows as decoration in which
case we also obtain a simple product formula. We are also able to provide
signless interpretations of our results. The proofs of the enumeration formulas
are based on a recent Littlewood-type identity, which allows us to reduce the
problem to the evaluations of two determinants. The evaluations are
accomplished by means of the LU-decompositions of the underlying matrices, and
an extension of Sister Celine's algorithm as well as creative telescoping to
evaluate certain triple sums. In particular, we use implementations of such
algorithms by Koutschan, and by Wegschaider and Riese.
