Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences

Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of P(x)^kQ(x), for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by a_K, we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum ∑^rp-1 _k=0 a_k, for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.