Sulanke, Robert A. A recurrence restricted by a diagonal condition: generalized Catalan arrays. (English) Zbl 0666.10008 Fibonacci Q. 27, No. 1, 33-46 (1989). Given a formal power series, \(f(x)\), and a nonnegative integer \(\mu\), the author investigates the series \(\Phi(x)\) satisfying \(\Phi(x)=f(x\Phi^{\mu}(x))\). He proves that \(\Phi(x)\) exists and is unique and derives a number of properties relating the coefficients of \(\Phi(x)\) to those of \(f(x)\). Applications include a derivation of Lagrange inversion and the enumeration of plane trees. Reviewer: David M. Bressoud (Saint Paul) Cited in 3 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 05A10 Factorials, binomial coefficients, combinatorial functions 05A15 Exact enumeration problems, generating functions Keywords:recurrence; diagonal series; Lagrange inversion; enumeration of plane trees PDF BibTeX XML Cite \textit{R. A. Sulanke}, Fibonacci Q. 27, No. 1, 33--46 (1989; Zbl 0666.10008) OpenURL Online Encyclopedia of Integer Sequences: The 4-Schroeder numbers: a(n) = number of lattice paths (Schroeder paths) from (0,0) to (3n,n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 3x.