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

Hankel determinants of middle binomial coefficients and conjectures for some polynomial extensions and modifications