I have a formal power series in one variable that I think might be algebraic (or perhaps just D-finite). Is there software that could help me explore this?
By way of comparison, there’s a very simple way to see if a formal power series appears to be rational: for small values of $n$, compute the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose entries are the first $2n+1$ coefficients of the formal power series. If the determinant is 0, then nontrivial elements of the nullspace correspond to possible $n$th order recurrence relations.
(I’m including the combinatorics tag since this sort of pattern-finding is sometimes an important early step in a combinatorial research project.)