Geodesic
The generating function of ternary trees and continued fractions
The electronic journal of combinatorics, Tome 13 (2006)

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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.
DOI : 10.37236/1079
Classification : 05A15, 05A10, 05A17, 30B70, 33C05
Mots-clés : Hankel determinants, plane partitions, alternating sign matrices, generating function, continued fraction, hypergeometric series, determinant identities 
Ira M. Gessel; Guoce Xin. The generating function of ternary trees and continued fractions. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1079
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     author = {Ira M. Gessel and Guoce Xin},
     title = {The generating function of ternary trees and continued fractions},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1079},
     zbl = {1098.05006},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1079/}
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