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A note on a one-parameter family of Catalan-like numbers. (English) Zbl 1201.11032

A number sequence \(a_n\) is Catalan-like if none of the Hankel determinants \(|a_{i+j}|_{i,j=0}^n\) is zero. For an integer parameter \(r\geq 1\) let \(a_n(r)\) be the coefficient of \(x^{n+1}\) in the power series expansion of the function obtained by reverting \[ \frac{x(1+rx)}{1+2rx+r(r+1)x^2}. \] The sequence \(a_n(r)\) is Catalan-like. The author derives properties of this family of sequences, including continued fraction expansions, associated orthogonal polynomials and associated Aigner matrices, which turn out to be Riordan arrays.

MSC:

11B83 Special sequences and polynomials
11Y55 Calculation of integer sequences
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

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