Taylor expansions for Catalan and Motzkin numbers

Sen Peng Eu, Shu Chung Liu, Yeong Nan Yeh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae associated with Catalan and Motzkin numbers. In particular, our expansion for Catalan number provides a simple proof of the classic Chung-Feller theorem; similar result for the Motzkin paths with flaws is also given.

Original languageEnglish
Pages (from-to)345-357
Number of pages13
JournalAdvances in Applied Mathematics
Issue number3
Publication statusPublished - 2002 Oct


  • Catalan and Motzkin numbers
  • Dyck and Motzkin paths
  • Taylor expansion

ASJC Scopus subject areas

  • Applied Mathematics


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