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We study a delay-differential analogue of the first Painlev\'e equationobtained as a delay periodic reduction of Shabat's dressing chain. We constructformal entire solutions to this equation and introduce a new family ofpolynomials (called Bernoulli-Catalan polynomials), which are defined by anonlinear recurrence of Catalan type, and which share properties with Bernoulliand Euler polynomials. We also discuss meromorphic solutions and describe thesingularity structure of this delay Painlev\'e-I equation in terms of an affineWeyl group of type $A_1^{(1)}$. As an application we demonstrate the link withthe problem of calculation of the Masur-Veech volumes of the moduli spaces ofmeromorphic quadratic differentials by re-deriving some of the known formulas.

Delay Painlev\'e-I equation, associated polynomials and Masur-Veech volumes