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{\bf \"Omer E\u{g}ecio\u{g}lu, Timothy Redmond and Charles Ryavec}
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{\bf Almost Product Evaluation of Hankel Determinants}
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An extensive literature exists describing various techniques for the
evaluation of Hankel determinants. The prevailing methods such as
Dodgson condensation, continued fraction expansion, LU decomposition,
all produce product formulas when they are applicable. We mention the
classic case of the Hankel determinants with binomial entries ${3 k +2
\choose k}$ and those with entries ${3 k \choose k}$; both of these
classes of Hankel determinants have product form evaluations.  The
intermediate case, ${3 k +1 \choose k}$ has not been evaluated.  There
is a good reason for this: these latter determinants do not have
product form evaluations.  In this paper we evaluate the Hankel
determinant of ${3 k +1 \choose k}$. The evaluation is a sum of a
small number of products, an {\em almost product}.  The method
actually provides more, and as applications, we present the salient
points for the evaluation of a number of other Hankel determinants
with polynomial entries, along with product and almost product form
evaluations at special points.



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