2021•arXiv (Cornell University)Open access
Hankel determinants of middle binomial coefficients and conjectures for\n some polynomial extensions and modifications
Johann Cigler
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Abstract
The middle binomial coefficients can be interpreted as numbers of Motzkin\npaths which have no horizontal steps at positive heights. Assigning suitable\nweights gives some nice polynomial extensions. We determine the Hankel\ndeterminants and their generating functions for the middle binomial\ncoefficients and derive many conjectures for their polynomial extensions.\nFinally, we explore experimentally some modifications of the middle binomial\ncoefficients whose Hankel determinants show an interesting modular pattern and\nobtain some q-analogs.\n
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