Some relatives of the Catalan sequence

We study a family of sequences $c_n(a_2,\ldots,a_r)$, where $r\ge2$ and $a_2,\ldots,a_r$ are real parameters. We find a sufficient condition for positive definiteness of the sequence $c_n(a_2,\ldots,a_r)$ and check several examples from OEIS. We also study relations of these sequences with the free and monotonic convolution.

[1]  Roland Speicher,et al.  Free Probability and Random Matrices , 2014, 1404.3393.

[2]  R. Speicher,et al.  Lectures on the Combinatorics of Free Probability: The free commutator , 2006 .

[3]  Naofumi Muraki,et al.  MONOTONIC INDEPENDENCE, MONOTONIC CENTRAL LIMIT THEOREM AND MONOTONIC LAW OF SMALL NUMBERS , 2001 .

[4]  Josef Hofbauer,et al.  Communicated by the Managing Editors , 1972 .

[5]  Alexandru Nica,et al.  Free random variables , 1992 .

[6]  A. Holland,et al.  Introduction to the theory of entire functions , 1973 .

[7]  David Callan,et al.  Another Type of Lattice Path: 10658 , 2000, Am. Math. Mon..

[8]  Thomas M. Richardson,et al.  The Super Patalan Numbers , 2014, J. Integer Seq..

[9]  Durham,et al.  ON GENERATING FUNCTIONS OF HAUSDORFF MOMENT SEQUENCES , 2014, 1401.8052.

[10]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[11]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

[12]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science (2. ed.) , 1994 .

[13]  K. Życzkowski,et al.  Densities of the Raney distributions , 2012, Documenta Mathematica.

[14]  W. Donoghue.  Monotone Matrix Functions and Analytic Continuation , 1974 .

[15]  Karol A. Penson,et al.  Coherent States from Combinatorial Sequences , 2001 .

[16]  W. Mlotkowski,et al.  PROBABILITY DISTRIBUTIONS WITH BINOMIAL MOMENTS , 2013, 1309.0595.

[17]  P. Forrester,et al.  Raney Distributions and Random Matrix Theory , 2014, 1404.5759.

[18]  W. Mlotkowski,et al.  Fuss-Catalan numbers in noncommutative probability , 2010, Documenta Mathematica.

[19]  A. Lauve,et al.  New Hopf Structures on Binary Trees , 2009 .

[20]  Karol Zyczkowski,et al.  Product of Ginibre matrices: Fuss-Catalan and Raney distributions. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Daniel Birmajer,et al.  Colored partitions of a convex polygon by noncrossing diagonals , 2015, Discret. Math..