Automated Discovery and Proof of Congruence Theorems for Partial Sums of Combinatorial Sequences

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 $\sum_{k=0}^{rp -1} a_k$,foranyprime$p$,andanypositiveinteger$r$, 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 $p$.