## 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