  A differentially recursive sequence over a differential field is a sequence
of elements satisfying a homogeneous differential equation with non-constant
coefficients (namely, Taylor expansions of elements of the field) in the
differential algebra of Hurwitz series. The main aim of this paper is to
explore the space of all differentially recursive sequences over a given field
with a non-zero differential. We show that these sequences form a two-sided
vector space that admits, in a canonical way, a structure of Hopf algebroid
over the subfield of constant elements. We prove that it is the direct limit,
as a left comodule, of all spaces of formal solutions of linear differential
equations and that it satisfies, as Hopf algebroid, an additional universal
property. When the differential on the base field is zero, we recover the Hopf
algebra structure of linearly recursive sequences.
