Dual coalgebra of the differential polynomial algebra in one variable and related coalgebras

We show that the dual coalgebra of the polynomial algebra in one variable is the space of linearly recursive sequences over an arbitrary field. Moreover, this coalgebra is a differential one relative to the dual standard derivation and does not contain nonzero finite-dimensional differentially closed subcoalgebras if the characteristic of the ground field is zero. We construct a Novikov coalgebra which is the dual coalgebra of the left-symmetric Witt algebra of index one. Also, we construct a Jordan supercoalgebra which is dual to the Jordan superalgebra of vector type of the polynomial algebra in one variable. All these coalgebras do not contain non-zero finite-dimensional subcoalgebras if the characteristic of ground field is zero. It is shown that over a field of characteristic different from 2 the adjoint Lie coalgebra of the dual coalgebra of the left-symmetric Witt algebra of index one is isomorphic to the dual coalgebra of the Witt algebra of index one.