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{\bf Ira M. Gessel and Guoce Xin}
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{\bf The Generating Function of Ternary Trees and Continued Fractions}
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Michael Somos conjectured a relation between Hankel determinants whose
entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number
of certain plane partitions and alternating sign matrices. Tamm
evaluated these determinants by showing that the generating function
for these entries has a continued fraction that is a special case of
Gauss's continued fraction for a quotient of hypergeometric series. We
give a systematic application of the continued fraction method to a
number of similar Hankel determinants. We also describe a simple
method for transforming determinants using the generating function for
their entries. In this way we transform Somos's Hankel determinants to
known determinants, and we obtain, up to a power of $3$, a Hankel
determinant for the number of alternating sign matrices. We obtain a
combinatorial proof, in terms of nonintersecting paths, of determinant
identities involving the number of ternary trees and more general
determinant identities involving the number of $r$-ary trees.
\bye