  The middle binomial coefficients can be interpreted as numbers of Motzkin
paths which have no horizontal steps at positive heights. Assigning suitable
weights gives some nice polynomial extensions. We determine the Hankel
determinants and their generating functions for the middle binomial
coefficients and derive many conjectures for their polynomial extensions.
Finally, we explore experimentally some modifications of the middle binomial
coefficients whose Hankel determinants show an interesting modular pattern and
obtain some q-analogs.
