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A recurrence restricted by a diagonal condition: generalized Catalan arrays. (English) Zbl 0666.10008

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.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions
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