Parametric Catalan Numbers and Catalan Triangles

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Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schroder numbers and small Schroder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.
Original languageAmerican English
JournalLinear Algebra and Its Applications
StatePublished - Feb 2013


  • Catalan number
  • large Schroder number
  • small Scroder number
  • Motzkin numbers
  • Riordan array
  • Bell-type Riordan array
  • characteristic sequence.


  • Applied Mathematics
  • Mathematics

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