  The ideal of the arc scheme of a double point or, equivalently, the
differential ideal generated by the ideal of a double point is a primary ideal
in an infinite-dimensional polynomial ring supported at the origin. This ideal
has a rich combinatorial structure connecting it to singularity theory,
partition identities, representation theory, and differential algebra. Macaulay
inverse system is a powerful tool for studying the structure of primary ideals
which describes an ideal in terms of certain linear differential operators. In
the present paper, we show that the inverse system of the ideal of the arc
scheme of a double point is precisely a vector space spanned by all the
Wronskians of the variables and their formal derivatives. We then apply this
characterization to extend our recent result on Poincar\'e-type series for such
ideals.
