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