Barry, Paul A note on a one-parameter family of Catalan-like numbers. (English) Zbl 1201.11032 J. Integer Seq. 12, No. 5, Article ID 09.5.4, 12 p. (2009). 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. Reviewer: Florin Nicolae (Berlin) Cited in 3 Documents MSC: 11B83 Special sequences and polynomials 11Y55 Calculation of integer sequences 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:integer sequence; Catalan-like; Riordan arrays; Hankel transform; orthogonal polynomials; Chebyshev polynomials Software:OEIS PDFBibTeX XMLCite \textit{P. Barry}, J. Integer Seq. 12, No. 5, Article ID 09.5.4, 12 p. (2009; Zbl 1201.11032) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: a(n) = binomial(n, floor(n/2)). Triangle read by rows: T(n,k) = Sum_{j>=0} j!*T(n-j-1, k-1) for n >= 0, k >= 0. Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0)} Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}. Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (1, 0), (1, 1)}. a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1. a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.