The examples involving Trinomial Coefficients and Catalan numbers (also see

here) are examples of P-recursive sequences, which is what I am generally after.

After doing a bit of research, I was able to find

this article, which explicitly states that the N-th term of a P-recursive sequence can be computed in no more than O(N^(1/2)) operations.

This paper also seems to agree.

Unfortunately, that's not practical for (primality) testing large numbers. That being said, it's not impossible to design an algorithm that can reduce the complexity from almost linear to polynomial time, as is the case of C-recursive sequences (Fibonacci, Lucas, Mersenne,..).

My idea is to reduce the baby-step giant step techniques with something analogous to square-multiply for P-recursive sequences, which I'm sure exists.

If anyone know of a paper that proves otherwise, or some better ideas, please let me know.