  Many combinatorial sequences (for example, the Catalan and Motzkin numbers)
may be expressed as the constant term of $P(x)^k Q(x)$, 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 $\sum_{k=0}^{rp -1}
a_k$, for {\it 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 $p$.

Combinatorics